1911 Encyclopædia Britannica/Magnetism, Terrestrial

by Charles Chree

MAGNETISM, TERRESTRIAL, the science which has for its province the study of the magnetic phenomena of the earth.

§ 1. Terrestrial magnetism has a long history. Its early growth was slow, and considerable uncertainty prevails as to its earliest developments. The properties of the magnet (see Magnetism) were to some small extent known to the Greeks and Romans before the Christian era, and compassesHistorical. (see Compass) of an elementary character seem to have been employed in Europe at least as early as the 12th century. In China and Japan compasses of a kind seem to have existed at a much earlier date, and it is even claimed that the Chinese were aware of the declination of the compass needle from the true north before the end of the 11th century. Early scientific knowledge was usually, however, a mixture of facts, very imperfectly ascertained, with philosophical imaginings. When an early writer makes a statement which to a modern reader suggests a knowledge of the declination of the compass, he may have had no such definite idea in his mind. So far as Western civilization is concerned, Columbus is usually credited with the discovery—in 1492 during his first voyage to America—that the pointing of the compass needle to the true north represents an exceptional state of matters, and that a declination in general exists, varying from place to place. The credit of these discoveries is not, however, universally conceded to Columbus. G. Hellmann ^{6 [n 1]} considers it almost certain that the departure of the needle from the true north was known in Europe before the time of Columbus. There is indirect evidence that the declination of the compass was not known in Europe in the early part of the 15th century, through the peculiarities shown by early maps believed to have been drawn solely by regard to the compass. Whether Columbus was the first to observe the declination or not, his date is at least approximately that of its discovery.

The next fundamental discovery is usually ascribed to Robert Norman, an English instrument maker. In The Newe Attractive (1581) Norman describes his discovery made some years before of the inclination or dip. The discovery was made more or less by accident, through Norman’s noticing that compass needles which were truly balanced so as to be horizontal when unmagnetized, ceased to be so after being stroked with a magnet. Norman devised a form of dip-circle, and found a value for the inclination in London which was at least not very wide of the mark.

Another fundamental discovery, that of the secular change of the declination, was made in England by Henry Gellibrand, professor of mathematics at Gresham College, who described it in his Discourse Mathematical on the Variation of the Magneticall Needle together with its Admirable Diminution lately discovered (1635). The history of this discovery affords a curious example of knowledge long delayed. William Borough, in his Discourse on the Variation of the Compas or Magneticall Needle (1581), gave for the declination at Limehouse in October 1580 the value 11°1/4 E. approximately. Observations were repeated at Limehouse, Gellibrand tells us, in 1622 by his colleague Edmund Gunter, professor of astronomy at Gresham College, who found the much smaller value 6° 13′. The difference seems to have been ascribed at first to error on Borough’s part, and no suspicion of the truth seems to have been felt until 1633, when some rough observations gave a value still lower than that found by Gunter. It was not until midsummer 1634 that Gellibrand felt sure of his facts, and yet the change of declination since 1580 exceeded 7°. The delay probably arose from the strength of the preconceived idea, apparently universally held, that the declination was absolutely fixed. This idea, it would appear, derived some of its strength from the positive assertion made on the point by Gilbert of Colchester in his De magnete (1600).

A third fundamental discovery, that of the diurnal change in the declination, is usually credited to George Graham (1675–1751), a London instrument maker. Previous observers, e.g. Gellibrand, had obtained slightly different values for the declination at different hours of the day, but it was natural to assign them to instrumental uncertainties. In those days the usual declination instrument was the compass with pivoted needles, and Graham himself at first assigned the differences he observed to friction. The observations on which he based his conclusions were made in 1722; an account of them was communicated to the Royal Society and published in the Philosophical Transactions for 1724.

The movements of the compass needle throughout the average day represent partly a regular diurnal variation, and partly irregular changes in the declination. The distinction, however, was not at first very clearly realized. Between 1756 and 1759 J. Canton observed the declination-changes on some 600 days, and was thus able to deduce their general character. He found that the most prominent part of the regular diurnal change in England consisted of a westerly movement of the north-pointing pole from 8 or 9 a.m. to 1 or 2 p.m., followed by a more leisurely return movement to the east. He also found that the amplitude of the movement was considerably larger in summer than in winter. Canton further observed that in a few days the movements were conspicuously irregular, and that aurora was then visible. This association of magnetic disturbance and aurora had, however, been observed somewhat before this time, a description of one conspicuous instance being contributed to the Royal Society in 1750 by Pehr Vilhelm Wargentin (1717–1783), a Swede.

Another landmark in the history of terrestrial magnetism was the discovery towards the end of the 18th century that the intensity of the resultant magnetic force varies at different parts of the earth. The first observations clearly showing this seem to be those of a Frenchman, Paul de Lamanon, who observed in 1785–1787 at Teneriffe and Macao, but his results were not published at the time. The first published observations seem to be those made by the great traveller Humboldt in tropical America between 1798 and 1803. The delay in this discovery may again be attributed to instrumental imperfections. The method first devised for comparing the force at different places consisted in taking the time of oscillation of the dipping needle, and even with modern circles this is hardly a method of high precision. Another discovery worth chronicling was made by Arago in 1827. From observations made at Paris he found that the inclination of the dipping needle and the intensity of the horizontal component of the magnetic force both possessed a diurnal variation.

§ 2. Whilst Italy, England and France claim most of the early observational discoveries, Germany deserves a large share of credit for the great improvement in instruments and methods during the first half of the 19th century. Measurements of the intensity of the magnetic force were somewhat crude until Gauss showed how absolute results could be obtained, and not merely relative data based on observations with some particular needle. Gauss also devised the bifilar magnetometer, which is still largely represented in instruments measuring changes of the horizontal force; but much of the practical success attending the application of his ideas to instruments seems due to Johann von Lamont (1805–1879), a Jesuit of Scottish origin resident in Germany.

The institution of special observatories for magnetic work is largely due to Humboldt and Gauss. The latter’s observatory at Göttingen, where regular observations began in 1834, was the centre of the Magnetic Union founded by Gauss and Weber for the carrying out of simultaneous magnetic observations and it was long customary to employ Göttingen time in schemes of international co-operation.

In the next decade, mainly through the influence of Sir Edward Sabine (1788–1883), afterwards president of the Royal Society, several magnetic observatories were established in the British colonies, at St Helena, Cape of Good Hope, Hobarton (now Hobart) and Toronto. These, with the exception of Toronto, continued in full action for only a few years; but their records—from their widely distributed positions—threw much fresh light on the differences between magnetic phenomena in different regions of the globe. The introduction of regular magnetic observatories led ere long to the discovery that there are notable differences between the amplitudes of the regular daily changes and the frequency of magnetic disturbances in different years. The discovery that magnetic phenomena have a period closely similar to, if not absolutely identical with, the “eleven year” period in sunspots, was made independently and nearly simultaneously about the middle of the 19th century by Lamont, Sabine and R. Wolf.

The last half of the 19th century showed a large increase in the number of observatories taking magnetic observations. After 1890 there was an increased interest in magnetic work. One of the contributory causes was the magnetic survey of the British Isles made by Sir A. Rücker and Sir T. E. Thorpe, which served as a stimulus to similar work elsewhere; another was the institution by L. A. Bauer of a magazine. Terrestrial Magnetism, specially devoted to the subject. This increased activity added largely to the stock of information, sometimes in forms of marked practical utility; it was also manifested in the publication of a number of papers of a speculative character. For historical details the writer is largely indebted to the works of E. Walker^[1] and L. A. Bauer.^[2]

§ 3. All the more important magnetic observatories are provided with instruments of two kinds. Those of the first kind give the absolute value of the magnetic elements at the time of observation. The unifilar magnetometer (q.v.), for instance, gives the absolute values of the declination and Observational Methods and Records. horizontal force, whilst the inclinometer (q.v.) or dip circle gives the inclination of the dipping needle. Instruments of the second kind, termed magnetographs (q.v.), are differential and self-recording, and show the changes constantly taking place in the magnetic elements. The ordinary form of magnetograph records photographically. Light reflected from a fixed mirror gives a base line answering to a constant value of the element in question; the light is cut off every hour or second hour so that the base line also serves to make the time. Light reflected from a mirror carried by a magnet gives a curved line answering to the changes in position of the magnet. The length of the ordinate or perpendicular drawn from any point of the curved line on to the base line is proportional to the extent of departure of the magnet from a standard position. If then we know the absolute value of the element which corresponds to the base line, and the equivalent of 1 cm. of ordinate, we can deduce the absolute value of the element answering to any given instant of time. In the case of the declination the value of 1 cm. of ordinate is usually dependent almost entirely on the distance of the mirror carried by the magnet from the photographic paper, and so remains invariable or very nearly so. In the case of the horizontal force and vertical force magnetographs—these being the two force components usually recorded—the value of 1 cm. of ordinate alters with the strength of the magnet. It has thus to be determined from time to time by observing the deflection shown on the photographic paper when an auxiliary magnet of known moment, at a measured distance, deflects the magnetograph magnet. Means are provided for altering the sensitiveness, for instance, by changing the effective distance in the bifilar suspension of the horizontal force magnet, and by altering the height of a small weight carried by the vertical force magnet. It is customary to aim at keeping the sensitiveness as constant as possible. A very common standard is to have 1 cm. of ordinate corresponding to 10′ of arc in the declination and to 50γ (1γ ≡ 0.00001 C.G.S.) in the horizontal and vertical force magnetographs.

As an example of how the curves are standardized, suppose that absolute observations of declination are taken four times a month, and that in a given month the mean of the observed values is 16° 34′.6 W. The curves are measured at the places which correspond to the times of the four observations, and the mean length of the four ordinates is, let us say, 2.52 cms. If 1 cm. answers to 10′, then 2.52 cms. represents 25′.2, and thus the value of the base line—i.e. the value which the declination would have if the curve came down to the base line—is for the month in question 16° 34′.6 less 25′.2 or 16° 9′.4. If now we wish to know the declination at any instant in this particular month all we have to do is to measure the corresponding ordinate and add its value, at the rate of 10′ per cm., to the base value 16° 9′.4 just found. Matters are a little more complicated in the case of the horizontal and vertical force magnetographs. Both instruments usually possess a sensible temperature coefficient, i.e. the position of the magnet is dependent to some extent on the temperature it happens to possess, and allowance has thus to be made for the difference from a standard temperature. In the case of the vertical force an “observed” value is derived by combining the observed value of the inclination with the simultaneous value of the horizontal force derived from the horizontal force magnetograph after the base value of the latter has been determined. In themselves the results of the absolute observations are of minor interest. Their main importance is that they provide the means of fixing the value of the base line in the curves. Unless they are made carefully and sufficiently often the information derivable from the curves suffers in accuracy, especially that relating to the secular change. It is from the curves that information is derived as to the regular diurnal variation and irregular changes. In some observatories it is customary to publish a complete record of the values of the magnetic elements at every hour for each day of the year. A useful and not unusual addition to this is a statement of the absolutely largest and smallest values of each element recorded during each day, with the precise times of their occurrence. On days of large disturbance even hourly readings give but a very imperfect idea of the phenomena, and it is customary at some observatories, e.g. Greenwich, to reproduce the more disturbed curves in the annual volume. In calculating the regular diurnal variation it is usual to consider each month separately. So far as is known at present, it is entirely or almost entirely a matter of accident at what precise hours specially high or low values of an element may present themselves during an individual highly disturbed day; whilst the range of the element on such a day may be 5, 10 or even 20 times as large as on the average undisturbed day of the month. It is thus customary when calculating diurnal inequalities to omit the days of largest disturbance, as their inclusion would introduce too large an element of uncertainty. Highly disturbed days are more than usually common in some years, and in some months of the year, thus their omission may produce effects other than that intended. Even on days of lesser disturbance difficulties present themselves. There may be to and fro movements of considerable amplitude occupying under an hour, and the hour may come exactly at the crest or at the very lowest part of the trough. Thus, if the reading represents in every case the ordinate at the precise hour a considerable element of chance may be introduced. If one is dealing with a mean from several hundred days such “accidents” can be trusted to practically neutralize one another, but this is much less fully the case when the period is as short as a month. To meet this difficulty it is customary at some observatories to derive hourly values from a freehand curve of continuous curvature, drawn so as to smooth out the apparently irregular movements. Instead of drawing a freehand curve it has been proposed to use a planimeter, and to accept as the hourly value of the ordinate the mean derived from a consideration of the area included between the curve, the base line and ordinates at the thirty minutes before and after each hour.

§ 4. Partly on account of the uncertainties due to disturbances, and partly with a view to economy of labour, it has been the practice at some observatories to derive diurnal inequalities from a comparatively small number of undisturbed or quiet days. Beginning with 1890, five days a month were selected at Greenwich by the astronomer royal as conspicuously quiet. In the selection regard was paid to the desirability that the arithmetic mean of the five dates should answer to near the middle of the month. In some of the other English observatories the routine measurement of the curves was limited to these selected quiet days. At Greenwich itself diurnal inequalities were derived regularly from the quiet days alone and also from all the days of the month, excluding those of large disturbance. If a quiet day differed from an ordinary day only in that the diurnal variation in the latter was partly obscured by irregular disturbances, then supposing enough days taken to smooth out irregularities, one would get the same diurnal inequality from ordinary and from quiet days. It was found, however, that this was hardly ever the case (see §§ 29 and 30). The quiet day scheme thus failed to secure exactly what was originally aimed at; on the other hand, it led to the discovery of a number of interesting results calculated to throw valuable sidelights on the phenomena of terrestrial magnetism.

The idea of selecting quiet days seems due originally to H. Wild. His selected quiet days for St Petersburg and Pavlovsk were very few in number, in some months not even a single day reaching his standard of freedom from disturbance. In later years the International Magnetic Committee requested the authorities of each observatory to arrange the days of each month in three groups representing the quiet, the moderately disturbed and the highly disturbed. The statistics are collected and published on behalf of the committee, the first to undertake the duty being M. Snellen. The days are in all cases counted from Greenwich midnight, so that the results are strictly synchronous. The results promise to be of much interest.

§ 5. The intensity and direction of the resultant magnetic force at a spot—i.e. the force experienced by a unit magnetic pole—are known if we know the three components of force parallel to any set of orthogonal axes. It is usual to take for these axes the vertical at the spot and two perpendicular axes in the horizontal plane; the latter are usually taken in and perpendicular to the geographical meridian. The usual notation in mathematical work is X to the north, Y to the west or east, and Z vertically downwards. The international magnetic committee have recommended that Y be taken positive to the east, but the fact that the declination is westerly over most of Europe has often led to the opposite procedure, and writers are not always as careful as they should be in stating their choice. Apart from mathematical calculations, the more usual course is to define the force by its horizontal and vertical components—usually termed H and V—and by the declination or angle which the horizontal component makes with the astronomical meridian. The declination is sometimes counted from 0° to 360°, 0° answering to the case when the so-called north pole (or north seeking pole) is directed towards geographical north, 90° to the case when it is directed to the east, and so on. It is more usual, however, to reckon declination only from 0° to 180°, characterizing it as easterly or westerly according as the north pole points to the east or to the west of the geographical meridian. The force is also completely defined by H or V, together with D the declination, and I the inclination to the horizon of the dipping needle. Instead of H and D some writers make use of N the northerly component, and W the westerly (or E the easterly). The resultant force itself is denoted sometimes by R, sometimes by T (total force). The following relationships exist between the symbols

X ≡ N, Y ≡ W or E, Z ≡ V, R ≡ T,
H ≡ √(X² + Y²), R ≡ √(X² + Y² + Z²),
tan D = Y / X, tan I = V / H.

The term magnetic element is applied to R or any of the components, and even to the angles D and I.

§ 6. Declination is the element concerning which our knowledge is most complete and most reliable. With a good unifilar magnetometer, at a fixed observatory distant from the magnetic poles, having a fixed mark of known azimuth, the observational uncertainty in a single Charts. observation should not exceed 0′.5 or at most 1′.0. It cannot be taken for granted that different unifilars, even by the best makers, will give absolutely identical values for the declination, but as a matter of fact the differences observed are usually very trifling. The chief source of uncertainty in the observation lies in the torsion of the suspension fibre, usually of silk or more rarely of phosphor bronze or other metal. A very stout suspension must be avoided at all cost, but the fibre must not be so thin as to have a considerable risk of breaking even in skilled hands. Near a magnetic pole the directive force on the declination magnet is reduced, and the effects of torsion are correspondingly increased. On the other hand, the regular and irregular changes of declination are much enhanced. If an observation consisting of four readings of declination occupies twelve minutes, the chances are that in this time the range at an English station will not exceed 1′, whereas at an arctic or antarctic station it will frequently exceed 10′. Much greater uncertainty thus attaches to declination results in the Arctic and Antarctic than to those in temperate latitudes. In the case of secular change data one important consideration is that the observations should be taken at an absolutely fixed spot, free from any artificial source of disturbance. In the case of many of the older observations of which records exist, the precise spot cannot be very exactly fixed, and not infrequently the site has become unsuitable through the erection of buildings not free from iron. Apart from buildings, much depends on whether the neighbourhood is free from basaltic and other magnetic rocks. If there are no local disturbances of this sort, a few yards difference is usually without appreciable influence, and even a few miles difference is of minor importance when one is calculating the mean secular change for a long period of years. When, however, local disturbances exist, even a few feet difference in the site may be important, and in the absence of positive knowledge to the contrary it is only prudent to act as if the site were disturbed. Near a magnetic pole the declination naturally changes very rapidly when one travels in the direction perpendicular to the lines of equal declination, so that the exact position of the site of observation is there of special importance.

The usual method of conveying information as to the value of the declination at different parts of the earth’s surface is to draw curves on a map—the so-called isogonals—such that at all points on any one curve the declination at a given specified epoch has the same value. The information being of special use to sailors, the preparation of magnetic charts has been largely the work of naval authorities—more especially of the hydrographic department of the British admiralty. The object of the admiralty world charts—four of which are reproduced here, on a reduced scale, by the kind permission of the Hydrographer—is rather to show the general features boldly than to indicate minute details. Apart from the immediate necessities of the case, this is a counsel of prudence. The observations used have mostly been taken at dates considerably anterior to that to which the chart is intended to apply. What the sailor wants is the declination now or for the next few years, not what it was five, ten or twenty years ago. Reliable secular change data, for reasons already indicated, are mainly obtainable from fixed observatories, and there are enormous areas outside of Europe where no such observatories exist. Again, as we shall see presently, the rate of the secular change sometimes alters greatly in the course of a comparatively few years. Thus, even when the observations themselves are thoroughly reliable, the prognostication made for a future date by even the most experienced of chart makers may be occasionally somewhat wide of the mark. Fig. 1 is a reduced copy of the British admiralty declination chart for the epoch 1907. It shows the isogonals between 70° N. and 65° S. latitude. Beyond the limits of this chart, the number of exact measurements of declination is somewhat limited, but the general nature of the phenomena is easily inferred. The geographical and the magnetic poles—where the dipping needle is vertical—are fundamental points. The north magnetic pole is situated in North America near the edge of the chart. We have no reason to suppose that the magnetic pole is really a fixed point, but for our present purpose we may regard it as such. Let us draw an imaginary circle round it, and let us travel round the circle in the direction, west, north, east, south, starting from a point where the north pole of a magnet (i.e. the pole which in Europe or the United States points to the north) is directed exactly towards the astronomical north. The point we start from is to the geographical south of the magnetic pole. As we go round the circle the needle keeps directed to the magnetic pole, and so points first slightly to the east of geographical north, then more and more to the east, then directly east, then to south of east, then to due south, to west of south, to west, to north-west, and finally when we get round to our original position due north once more. Thus, during our course round the circle the needle will have pointed in all possible directions. In other words, isogonals answering to all possible values of the declination have their origin in the north magnetic pole. The same remark applies of course to the south magnetic pole.

Fig. 1.—Isogonals, or lines of equal magnetic declination.

Now, suppose ourselves at the north geographical pole of the earth. Neglecting as before diurnal variation and similar temporary changes, and assuming no abnormal local disturbance, the compass needle at and very close to this pole will occupy a fixed direction relative to the ground underneath. Let us draw on the ground through the pole a straight line parallel to the direction taken there by the compass needle, and let us carry a compass needle round a small circle whose centre is the pole. At all points on the circle the positions of the needle will be parallel; but whereas the north pole of the magnet will point exactly towards the centre of the circle at one of the points where the straight line drawn on the ground cuts the circumference, it will at the opposite end of the diameter point exactly away from the centre. The former part is clearly on the isogonal where the declination is 0°, the latter on the isogonal where it is 180°. Isogonals will thus radiate out from the north geographical pole (and similarly of course from the south geographical pole) in all directions. If we travel along an isogonal, starting from the north magnetic pole, our course will generally take us, often very circuitously, to the north geographical pole. If, for example, we select the isogonal of 10° E., we at first travel nearly south, but then more and more westerly, then north-westerly across the north-east of Asia; the direction then gets less northerly, and makes a dip to the south before finally making for the north geographical pole. It is possible, however, according to the chart, to travel direct from the north magnetic to the south geographical pole, provided we select an isogonal answering to a small westerly or easterly declination (from about 19° W. to 7° E.).

Special interest attaches to the isogonals answering to declination 0°. These are termed agonic lines, but sailors often call them lines of no variation, the term variation having at one time been in common use in the sense of declination. If we start from the north magnetic pole the agonic line takes us across Canada, the United States and South America in a fairly straight course to the south geographical pole. A curve continuous with this can be drawn from the south geographical to the south magnetic pole at every point of which the needle points in the geographical meridian; but here the north pole of the needle is pointing south, not north, so that this portion of curve is really an isogonal of 180°. In continuation of this there emanates from the south magnetic pole a second isogonal of 0°, or agonic line, which traverses Australia, Arabia and Russia, and takes us to the north geographical pole. Finally, we have an isogonal of 180°, continuous with this second isogonal of 0° which takes us to the north magnetic pole, from which we started. Throughout the whole area included within these isogonals of 0° and 180°—excluding locally disturbed areas—the declination is westerly; outside this area the declination is in general easterly. There is, however, as shown in the chart, an isogonal of 0° enclosing an area in eastern Asia inside which the declination is westerly though small.

§ 7. Fig. 2 is a reduced copy of the admiralty chart of inclination or dip for the epoch 1907. The places where the dip has the same value lie on curves called isoclinals. The dip is northerly (north pole dips) or southerly (south pole dips) according as the place is north or south of the isoclinal of 0°. At places actually on this isoclinal the dipping needle is horizontal. The isoclinal of 0° is nowhere very far from the geographical equator, but lies to the north of it in Asia and Africa, and to the south of it in South America. As we travel north from the isoclinal of 0° along the meridian containing the magnetic pole the dipping needle’s north pole dips more and more, until when we reach the magnetic pole the needle is vertical. Going still farther north, we have the dip diminishing. The northerly inclination is considerably less in Europe than in the same latitudes of North America; and correspondingly the southerly inclination is less in South America than in the same latitudes of Africa.

Fig. 2.—Isoclinals, or lines of equal magnetic dip.

Fig. 3 is a reduced copy of the admiralty horizontal force chart for 1907. The curves, called isomagnetics, connect the places where the horizontal force has the same value; the force is expressed in C.G.S. units. The horizontal force vanishes of course at the magnetic poles. The chart shows a maximum value of between 0.39 and 0.40 in an oval including the south of Siam and the China Sea. The horizontal force is smaller in North America than in corresponding latitudes in Europe.

Fig. 3.—Isomagnetics, lines of equal horizontal force.

Charts are sometimes drawn for other magnetic elements, especially vertical force (fig. 4) and total force. The isomagnetic of zero vertical force coincides necessarily with that of zero dip, and there is in general considerable resemblance between the forms of lines of equal vertical force and those of equal dip. The highest values of the vertical force occur in areas surrounding the magnetic poles, and are fully 50% larger than the largest values of the horizontal force. The total force is least in equatorial regions, where values slightly under 0.4 C.G.S. are encountered. In the northern hemisphere there are two distinct maxima of total force. One of these so-called foci is in Canada, the other in the north-east of Siberia, the former having the higher value of the force. There are, however, higher values of the total force than at either of these foci throughout a considerable area to the south of Australia. In the northern hemisphere the lines of equal total force—called isodynamic lines—form two sets more or less distinct, consisting of closed ovals, one set surrounding the Canadian the other the Siberian focus.

§ 8. As already explained, magnetic charts for the world or for large areas give only a general idea of the values of the elements. If the region is undisturbed, very fairly approximate values are derivable from the charts, but when the highest accuracy is necessary the only thing to do is to observe at the precise spot. In disturbed areas local values often depart somewhat widely from what one would infer from the chart, and occasionally there are large differences between places only a few miles apart. Magnetic observatories usually publish the mean value for the year of their magnetic Magnetic Elements and their Secular Change. elements. It has been customary for many years to collect and publish these results in the annual report of the Kew Observatory (Observatory Department of the National Physical Laboratory). The data in Tables I. and II. are mainly derived from this source. The observatories are arranged in order of latitude, and their geographical co-ordinates are given in Table II., longitude being reckoned from Greenwich. Table I. gives the mean values of the declination, inclination and horizontal force for January 1, 1901; they are in the main arithmetic means of the mean annual values for the two years 1900 and 1901. The mean annual secular changes given in this table are derived from a short period of years—usually 1898 to 1903—the centre of which fell at the beginning of 1901. Table II. is similar to Table I., but includes vertical force results; it is more extensive and contains more recent data. In it the number of years is specified from which the mean secular change is derived; in all cases the last year of the period employed was that to which the absolute values assigned to the element belong. The great majority of the stations have declination west and inclination north; it has thus been convenient to attach the + sign to increasing westerly (or decreasing easterly) declination and to increasing northerly (or decreasing southerly) inclination. In other words, in the case of the declination + means that the north end of the needle is moving to the west, while in the case of the inclination + means that the north end (whether the dipping end or not) is moving towards the nadir. In the case, however, of the vertical force + means simply numerical increase, irrespective of whether the north or the south pole dips. The unit employed in the horizontal and vertical force secular changes is 1γ, i.e. 0.00001 C.G.S. Even in the declination, at the very best observatories, it is hardly safe to assume that the apparent change from one year to the next is absolutely truthful to nature. This is especially the case if there has been any change of instrument or observer, or if any alteration has been made to buildings in the immediate vicinity. A change of instrument is a much greater source of uncertainty in the case of horizontal force or dip than in the case of declination, and dip circles and needles are more liable to deterioration than magnetometers. Thus, secular change data for inclination and vertical force are the least reliable. The uncertainties, of course, are much less, from a purely mathematical standpoint, for secular changes representing a mean from five or ten years than for those derived from successive years’ values of the elements. The longer, however, the period of years, the greater is the chance that one of the elements may in the course of it have passed through a maximum or minimum value. This possibility should always be borne in mind in cases where a mean secular change appears exceptionally small.

Fig. 4.—Isomagnetics, lines of equal vertical force.

As Tables I. and II. show, the declination needle is moving to the east all over Europe, and the rate at which it is moving seems not to vary much throughout the continent. The needle is also moving to the east throughout the western parts of Asia, the north and east of Africa, and the east of North America. It is moving to the west in the west of North America, in South America, and in the south and east of Asia, including Japan, south-east Siberia, eastern China and most of India.

§ 9. The information in figs. 1, 2, 3 and 4 and in Tables I. and II. applies only to recent years. Owing to secular change, recent charts differ widely from the earliest ones constructed. The first charts believed to have been constructed were those of Edmund Halley the astronomer. According to L. A. Bauer,^[3] who has made a special study of the subject, Halley issued two declination charts for the epoch 1700; one, published in 1701, was practically confined to the Atlantic Ocean, whilst the second, published in 1702, contained also data for the Indian Ocean and part of the Pacific. These charts showed the isogonic lines, but only over the ocean areas. Though the charts for 1700 were the first published, there are others which apply to earlier epochs. W. van Bemmelen^[4] has published charts for the epochs 1500, 1550, 1600, 1650 and 1700, whilst H. Fritsche^[5] has more recently published charts of declination, inclination and horizontal force for 1600, 1700, 1780, 1842 and 1915. A number of early declination charts were given in Hansteen’s Atlas and in G. Hellmann’s reprints. Die Altesten Karten der Isogonen, Isoklinen, Isodynamen (Berlin, 1895). The data for the earlier epochs, especially those prior to 1700, are meagre, and in many cases probably of indifferent accuracy, so that the reliability of the charts for these epochs is somewhat open to doubt.

If we take either Hansteen’s or Fritsche’s declination chart for 1600 we notice a profound difference from fig. 1. In 1600 the agonic line starting from the north magnetic pole, after finding its way south to the Gulf of Mexico, doubled back to the north-east, and passed across or near Iceland. After getting well to the north of Iceland it doubled again to the south, passing to the east of the Baltic. The second agonic line which now lies to the west of St Petersburg appears in 1600 to have continued, after traversing Australia, in a nearly northerly direction through the extreme east of China. The nature of the changes in declination in western Europe will be understood from Table III., the data from which, though derived from a variety of places in the south-east of England,^[6] may be regarded as approximately true of London. The earliest result is that obtained by Borough at Limehouse. Those made in the 16th century are due to Gunter, Gellibrand, Henry Bond and Halley. The observations from 1787 to 1805 were due to George Gilpin, who published particulars of his own and the earlier observations in the Phil. Trans. for 1806. The data for 1817 and 1820 were obtained by Col. Mark Beaufoy, at Bushey, Herts. They seem to come precisely at the time when the needle, which had been continuously moving to the west since the earliest observations, began to retrace

its steps. The data from 1860 onwards apply to Kew.

Table I.—Magnetic Elements and their Rate of Secular Change for January 1, 1901.

Place.	Absolute values.			Secular change.
	D.	I.	H.	D.	I.	H.
	° ′	° ′		′	′	γ
Pavlovsk	0 39.8E	70 36.8N	.16553	− 4.1	−0.8	+ 7
Ekatarinburg	10   6.3E	70 40.5N	.17783	− 4.6	+0.5	−13
Copenhagen	10 10.4W	68 38.5N	.17525
Stonyhurst	18 10.3W	68 48.0N	.17330	− 4.0		+22
Wilhelmshaven	12 26.0W	67 39.7N	.18108	− 4.1	−2.1	+20
Potsdam	9 54.2W	66 24.5N	.18852	− 4.2	−1.6	+16
Irkutsk	2   1.0E	70 15.8N	.20122	+ 0.5	+1.6	−14
de Bilt	13 48.3W	66 55.5N	.18516	− 4.4	−2.2	+14
Kew	16 50.8W	67 10.6N	.18440	− 4.2	−2.2	+25
Greenwich	16 27.5W	67  7.3N	.18465	− 4.0	−2.2	+23
Uccle	14 11.0W	66  8.8N	.18954	− 4.2	−2.1	+23
Falmouth	18 27.3W	66 44.0N	.18705	− 3.8	−2.7	+26
Prague	9  4.4W		.19956	− 4.4		+20
St Helier	16 58.1W	65 44.1N		− 3.5	−2.7
Parc St Maur	14 43.4W	64 52.3N	.19755	− 4.0	−2.2	+23
Val Joyeux	15 13.7W	65  0.0N	.19670
Munich	10 25.8W	63 18.1N	.20629	− 4.8	−2.7	+21
O’Gyalla	7 26.1W		.21164	− 4.8		+13
Pola	9 22.7W	60 14.5N	.22216	− 4.0		+23
Toulouse	14 16.4W	60 55.9N	.21945	− 3.9	−2.5	+25
Perpignan	13 34.7W	59 57.6N	.22453
Capo di Monte	9  8.0W	56 22.3N		− 5.2	−2.3
Madrid	15 39.0W
Coimbra	17 18.1W	59 22.0N	.22786	− 3.7	−4.3	+34
Lisbon	17 15.7W	57 53.0N	.23548
Athens	5 38.2W	52  7.5N	.26076
San Fernando	15 57.5W	55  8.8N	.24648
Tokyo	4 34.9W	49  0.3N	.29932
Zi−ka−wei	2 23.5W	45 43.5N	.32875	+ 1.5	−1.5	+37
Helwan	3 39.7W	40 30.8N	.30136	− 7.0	−0.4	− 7
Hong−Kong	0 17.5E	31 22.8N	.36753	+ 1.8	−4.3	+45
Kolaba	0 23.2E	21 26.5N	.37436	+ 2.2	+7.0	− 9
Manila	0 52.2E	16 13.5N	.38064	+ 0.1	−5.3	+47
Batavia	1   7.3E	30 35.5S	.36724	+ 3.0	−7.3	−11
Mauritius	9 25.2W	54  9.4S	.23820	− 4.7	+4.6	−39
Rio de Janeiro	8  2.9W	13 20.1S	.2501	+10.4	−2.3
Melbourne	8 25.6E	67 24.6S	.23295

The rate of movement of the needle to the east at London—and throughout Europe generally—fell off markedly subsequent to 1880. The change of declination in fact between 1880 and 1895 was only about 75% of that between 1865 and 1880, and the mean annual change from 1895 to 1900 was less than 75% of the mean annual change of the preceding fifteen years. Thus in 1902 it was at least open to doubt whether a change in the sign of the secular change were not in immediate prospect. Subsequent, however, to that date there was little further decline in the rate of secular change, and since 1905 there has been very distinct acceleration. Thus, if we derive a mean value from the eighteen European stations for which declination secular changes are given in Tables I. and II. we find

mean value from table	I.	−4.18
”   ”   ”   ”	II.	−5.21

The epoch to which the data in Table II. refer is somewhat variable, but is in all cases more recent than the epoch, January 1, 1901, for Table I., the mean difference being about 5 years.

§ 10. At Paris there seems to have been a maximum of easterly declination (about 9°) about 1580; the needle pointed to true north about 1662, and reached its extreme westerly position between 1812 and 1814. The phenomena at Rome resembled those at Paris and London, but the extreme westerly position is believed to have been attained earlier. The rate of change near the turning point seems to have been very slow, and as no fixed observatories existed in those days, the precise time of its occurrence is open to some doubt.

Perhaps the most complete observations extant as to the declination phenomena near a turning point relate to Kolaba observatory at Bombay; they were given originally by N. A. F. Moos,^[7] the director of the observatory. Some of the more interesting details are given in Table IV.; here W denotes movement to be west, and so answers to a numerical diminution in the declination, which is easterly.

Prior to 1880 the secular change at Kolaba was unmistakably to the east, and subsequent to 1883 it was clearly to the west; but between these dates opinions will probably differ as to what actually happened. The fluctuations then apparent in the sign of the annual change may be real, but it is at least conceivable that they are of instrumental origin. From 1870 to 1875 the mean annual change was −1′.2; from 1885 to 1890 it was +1′.5, from 1890 to 1895 it was +2′.0, while from 1895 to 1905 it was +2′.35, the + sign denoting movement to the west. Thus, in this case the rate of secular change has increased fairly steadily since the turning point was reached.

Table V. contains some data for St Helena and the Cape of Good Hope,^[8] both places having a long magnetic history. The remarkable feature at St Helena is the uniformity in the rate of secular change. The figures for the Cape show a reversal in the direction of the secular change about 1840, but after a few years the arrested movement to the west again became visible. According, however, to J. C. Beattie’s Magnetic Survey of South Africa the movement to the west ceased shortly after 1870. A persistent movement to the east then set in, the mean annual change increasing from 1′.8 between 1873 and 1890 to 3′.8 between 1890 and 1900.

§ 11. Secular changes of declination have been particularly interesting in the United States, an area about which information is unusually complete, thanks to the labours and publications of the United States Coast and Geodetic Survey.^[9] At present the agonic line passes in a south-easterly direction from Lake Superior to South Carolina. To the east of the agonic line the declination is westerly, and to the west it is easterly. In 1905 the declination varied from about 21° W. in the extreme north-east to about 24° E. in the extreme north-west. At present the motion of the agonic line seems to be towards the west, but it is very slow. To the east of the agonic line westerly declination is increasing, and to the west of the line, with the exception of a narrow strip immediately adjacent to it, easterly declination is increasing. The phenomena in short suggest a motion southwards in the north magnetic pole. Since 1750 declination has always been westerly in the extreme east of the States, and always easterly in the extreme west, but the position of the agonic line has altered a good deal. It was to the west of Richmond, Virginia, from 1750 to about 1772, then to the east of it until about 1838 when it once more passed to the west; since that time it has travelled farther to the west. Table VI. is intended to show the nature of the secular change throughout the whole country. As before, + denotes that the north pole of the magnet is moving to the west,—that it is moving to the east.

The data in Table VI. represent the mean change of declination per annum, derived from the period (ten years, except for 1900–1905) which ended in the year put at the top of the column. The stations are arranged in four groups, the first group representing the extreme eastern, the last group the extreme western states, the other two groups being intermediate. In each group the stations are arranged, at least approximately, in order of latitude. The data are derived from the values of the declination given in the Geodetic Survey’s Report for 1906, appendix 4, and Magnetic Tables and Magnetic Charts by L. A. Bauer, 1908. The values seem, in most cases, based to some extent on calculation, and very probably the secular change was not in reality quite so regular as the figures suggest. For the Western States the earliest data are comparatively recent, but for some of the eastern states data earlier than any in the table appear in the Report of the Coast and Geodetic Survey for 1902. These data indicate that the easterly movement of the magnet, visible in all the earlier figures for the Eastern States in Table VI., existed in all of them at least as far back as 1700. There is not very much evidence as to the secular change between 1700 and 1650, the earliest date to which the Coast and Geodetic Survey’s figures refer. The figures show a maximum of westerly declination about 1670 in New Jersey and about 1675 in Maryland. They suggest that this maximum was experienced all along the Atlantic border some time in the 17th century, but earlier in the extreme north-east than in New York or Maryland.

Examination of Table VI. shows that the needle continued to move to the east for some time after 1750 even in the Eastern States. But the rate of movement was clearly diminishing, and about 1765 the extreme easterly position was reached in Eastport, Maine, the needle then beginning to retrace its steps to the west. The phenomena visible at Maine are seen repeating themselves at places more and more to the west, in Boston about 1785, in Albany about 1800, in Washington, D.C., about 1805, in Columbus (Ohio) about 1815, in Montgomery (Alabama) about 1825, in Bloomington (Ill.) about 1830, in Des Moines (Iowa) about 1840, in Santa Rosa (New Mexico) about 1860 and in Salt Lake about 1870. In 1885 the needle was moving to the west over the whole United States with the exception of a comparatively narrow strip along the Pacific coast. Even an acute observer would have been tempted to prophesy in 1885 that at no distant date the secular change would be pronouncedly westerly right up to the Pacific. But in a few years a complete change took place. The movement to the east, which had become exceedingly small, if existent, in the Pacific states, began to accelerate; the movement to the west continued in the central, as in the eastern states, but perceptibly slackened. In 1905 the area throughout which the movement to the west still continued had greatly contracted and lay to the east of a line drawn from the west end of Lake Superior to the west of Georgia. If we take a station like Little Rock (Arkansas), we have the secular change to the west lasting for about sixty years. Further west the period shortens. At Pueblo (Colorado) it is about forty years, at Salt Lake under thirty years, at Prescott (Arizona) about twenty years. Considering how fast the area throughout which the secular change is easterly has extended to the east since 1885, one would be tempted to infer that at no distant date it will include the whole of the United States. In the extreme north-east, however, the movement of the needle to the west, which had slackened perceptibly after 1860 or 1870, is once more accelerating. Thus the auspices do not all

point one way, and the future is as uncertain as it is interesting.

Table II.—Recent Values of the Magnetic Elements and their Rate of Secular Change.

Place.	Geographical position.		Absolute Values of Elements.					Secular change (mean per annum).
	Latitude.	Longitude.	Year.	D.	I.	H.	V.	Interval in years.	D.	I.	H.	V.
	°  ′	°  ′		°  ′	°   ′				′	′
Pavlovsk	59 41N	30 29E	1906	1  4.2E	70 36.6N	.16528	.46963	5	−4.5	+0.1	− 6	−14
Sitka (Alaska)	57  3N	135 20W	1906	30  3.3E	74 41.7N	.15502	.56646	4	−3.0	−1.6	+18	−38
Ekatarinburg	56 49N	60 38E	1906	10 31.0E	70 49.5N	.17664	.50796	5	−4.5	+1.7	−23	+18
Rude Skov (Copenhagen)	55 51N	12 27E	1908	9 43.3W	68 45N	.17406	.44759
Stonyhurst	53 51N	2 28W	1909	17 28.6W	68 42.8N	.17424	.44722	5	−5.9	−1.1	+ 6	−25
Hamburg	53 33N	9 59E	1903	11 10.2W	67 23.5N	.18126	.43527
Wilhelmshaven	53 32N	8  9E	1909	11 46.8W		.18129		5	−5.2		− 7
Potsdam	52 23N	13  4E	1909	9 10.6W	66 20.0N	.18834	.42971	5	−5.8	+0.1	− 9	−19
Irkutsk	52 16N	104 16E	1905	1 58.1E	70 25.0N	.20011	.56250	5	+0.6	+2.0	−24	+39
de Bilt	52  5N	5 11E	1907	13 19.0W	66 49.9N	.18559	.43368	5	−4.7	−0.6	+ 2	−16
Valencia	51 56N	10 15W	1909	20 50.3W	68 15.1N	.17877	.44812	5	−5.0	−1.2	+ 7	−25
Kew	51 28N	0 19W	1909	16 10.8W	66 59.7N	.18506	.43588	5	−5.4	−1.1	+ 2	−35
Greenwich	51 28N	0  0	1909	15 47.6W	66 53.9N	.18526	.43432	5	−5.5	−0.7	+ 1	−20
Uccle	50 48N	4 21E	1908	13 36.7W	66  1.6N	.19061	.42867	4	−5.3	−0.8	− 3	−35
Falmouth	50  9N	5  5W	1909	17 48.4W	66 30.6N	.18802	.43266	5	−4.7	−1.4	+ 9	−30
Prague	50  5N	14 25E	1908	8 20.9W				5	−6.5
Cracow	50  4N	19 58E	1909	5 35.1W	64 18N			3	−7.3
St Helier	49 12N	2 5W	1907	16 27.4W	65 34.5N			5	−5.3	−1.2
Val Joyeux	48 49N	2 1E	1909	14 32.9W	64 43.9N	.19727	.41792	5	−5.4	−1.7	+ 1	−51
Vienna	48 15N	16 21E	1898	8 24.1W
Munich	48  9N	11 37E	1906	9 59.5W	63 10.0N	.20657	.40835	5	−4.8	−1.3	+ 4	−31
O’Gyalla	47 53N	18 12E	1909	6 43.9W		.21094		5	−5.0		−10
Odessa	46 26N	30 46E	1899	4 36.7W	62 18.2N	.21869	.41660
Pola	44 52N	15 51E	1908	8 43.2W	60  6.8N	.22207	.38640	5	−5.5	−0.6	− 4	−23
Agincourt (Toronto)	43 47N	79 16W	1906	5 45.3W	74 35.6N	.16397	.59502	4	+3.4	+0.9	−23	−24
Nice	43 43N	7 16E	1899	12  4.0W	60 11.7N	.22390	.39087
Toulouse	43 37N	1 28E	1905	13 56.3W	60 49.1N	.22025	.39439	5	−4.5	−1.5	+ 2	− 2
Perpignan	42 42N	2 53E	1907	13 4.4W				7	−4.7
Tiflis	41 43N	44 48E	1905	2 41.6E	56 2.8N	.25451	.37799	7	−5.2	+1.7	−26	+ 2
Capo di Monte	40 52N	14 15E	1906	8 40.3W	56 13.5N			5	−5.1	−1.5
Madrid	40 25N	3 40W	1901	15 35.6W
Coimbra	40 12N	8 25W	1908	16 46.2W	58 57.3N	.22946	.38120	5	−4.6	−2.9	+17	−45
Baldwin (Kansas)	38 47N	95 10W	1906	8 30.1E	68 45.1N	.21807	.56081	4	−1.7	+1.8	−36	− 8
Cheltenham(Maryland)	38 44N	76 50W	1906	5 22.0W	70 27.3N	.20035	.56436	4	+3.8	+1.2	−38	−45
Lisbon	38 43N	9  9W	1900	17 18.0W	57 54.8N	.23516	.37484
Athens	37 58N	21 23E	1908	4 52.9W	52 11.7N	.26197	.33613	5	−5.5
San Fernando	36 28N	6 12W	1908	15 25.6W	54 48.4N	.24829	.35206	5	−4.6	−2.8	+26	−24
Tokyo	35 41N	139 45E	1901	4 36.1W	49  0.0N	.29954	.34459
Zi-ka-wei	31 12N	121 26E	1906	2 32.0W	45 35.3N	.33040	.33726	5	+1.5	−1.3	+30	+ 6
Dehra Dun	30 19N	78  3E	1907	2 38.3E	43 36.1N	.33324	.31736	4	+0.8	+5.5	−26	+77
Helwan	29 52N	31 21E	1909	2 49.2W	40 40.4N	.30031	.25804	5	−5.7	+1.2	− 6	+13
Havana	23  8N	82 25W	1905	2 25.0E	52 57.4N	.30531	.40452
Barrackpore	22 46N	88 22E	1907	1  9.9E	30 30.2N	.37288	.21967	3	+4.2	+3.4	+21	+62
Hong-Kong	22 18N	114 10E	1908	0  3.9E	31  2.5N	.37047	.22292	5	+1.9	−1.8	+43	− 1
Honolulu	21 19N	158  4W	1906	9 21.7E	40  1.8N	.29220	.24545	4	−0.9	−3.2	−19	−62
Kolaba	18 54N	72 49E	1905	0 14.0E	21 58.5N	.37382	.15084	5	+2.1	+7.2	−11	+86
Alibagh	18 39N	72 52E	1909	1  0.3E	23 29.0N	.36845	.16008	3	+1.7	+6.8	−10	+82
Vieques (Porto Rico)	18  9N	65 26W	1906	1 33.2W	49 47.7N	.28927	.34224	2	+7.2	+6.8	−49	+66
Manila	14 35N	120 59E	1904	0 51.4E	16  0.2N	.38215	.10960	5	+0.1	−3.9	+47	−34
Kodaikanal	10 14N	77 28E	1907	0 40.7W	3 27.2N	.37431	.02259	4	+4.3	+5.5	+16	+61
Batavia	6 11S	106 49E	1906	0 54.1E	30 48.5S	.36708	.21889	4	+2.1	−7.7	− 2	+110
Dar es Salaam	6 49S	39 18E	1903	7 35.2W
Mauritius	20  6S	57 33E	1908	9 14.3W	53 44.9S	.23415	.31932	5	−0.3	+2.9	−53	−131
Rio de Janeiro	22 55S	43 11W	1906	8 55.5W	13 57.1S	.24772	.06164	5	+9.1	−6.8	−42	+44
Santiago (Chile)	33 27S	70 42W	1906	14 18.7E	30 11.8S			3	+6.1	+9.9
Melbourne	37 50S	144 58E	1901	8 26.7E	67 25.0S	.23305	.56024
Christchurch, N.Z.	43 32S	172 37E	1903	16 18.4E	67 42.3S	.22657	.55259

Table III.—Declination at London.

Date.	Declination.	Date.	Declination.	Date.	Declination.
	°   ′		°   ′		°   ′
1580	11   15E	1773	21   9W	1860	21   38.9W
1622	6    0	1787	23   19	1865	20   58.7
1634	4    6	1795	23   57	1870	20   18.3
1657	0    0	1802	24    6	1875	19   35.6
1665	1   22W	1805	24    8	1880	18   52.1
1672	2   30	1817	24   36	1885	18   19.2
1692	6    0	1818	24   38	1890	17   50.6
1723	14   17	1819	24   36	1895	17   16.8
1748	17   40	1820	24   34	1900	16   52.7
				1905	16   32.9

§ 12. Table VII. gives particulars of the secular change of horizontal force and northerly inclination at London. Prior to the middle of the 19th century information as to the value of H is of uncertain value. The earlier inclination data^[10] are due to Norman, Gilbert, Bond, Graham, Heberden and Gilpin. The data from 1857 onwards, both for H and I, refer to Kew. “London” is rather a vague term, but the differences between the values of H and I at Kew and Greenwich—in the extreme west and east—are almost nil. For some time after its discovery by Robert Norman inclination at London increased. The earlier observations are not sufficient to admit of the date of the maximum inclination or its absolute value being determined with precision. Probably the date was near 1723. This view is supported by the fact that at Paris the inclination fell from 72° 15′ in 1754 to 71° 48′ in 1780. The earlier observations in London were probably of no very high accuracy, and the rates of secular change deducible from them are correspondingly uncertain. It is not improbable that the average annual change 0′.8 derived from the thirteen years 1773–1786 is too small, and the value 6′.2 derived from the fifteen years 1786–1801 too large. There is, however, other evidence of unusually rapid secular change of inclination towards the end of the 18th century in western Europe; for observations in Paris show a fall of 56′ between 1780 and 1791, and of 90′ between 1791 and 1806. Between 1801 and 1901 inclination in London diminished by 3° 26′.5, or on the average by 2′.1 per annum, while between 1857 and 1900 H increased on the average by 22γ a year. These values differ but little from the secular changes given in Table I. as applying at Kew for the epoch Jan. 1, 1901. Since the beginning, however, of the 20th century a notable change has set in, which seems shared by the whole of western Europe. This is shown in a striking fashion by contrasting the data from European stations in Tables I. and II. There are fifteen of these stations which give secular change data for H in both tables, while thirteen give secular data for I. The mean values of the secular changes derived from these stations are as follows:—

	I	H
From Table I.	−2′.35	+21.0γ
From Table II.	−1.12	+1.6γ

The difference in epoch between the two sets of results is only about 5 years, and yet in that short time the mean rate of annual increase in H fell to a thirteenth of its original value. During 1908–1909 H diminished throughout all Europe except in the extreme west. Whether we have to do with merely a temporary phase, or whether a general and persistent diminution in the value of H is about to set in over Europe it is yet hardly possible to say.

Table IV.—Declination at Kolaba (Bombay).

Year.	Declination  East.	Change since  previous year.	Year.	Declination  East.	Change since  previous year.
	°   ′   ″	′   ″		°   ′   ″	′   ″
1876	0 55 58	0 37 E	1881	0 57 12	0  3 E
1877	56 39	0 41 E	1882	0 56 50	0 22 W
1878	57  6	0 27 E	1883	57  2	0 12 E
1879	57 30	0 24 E	1884	55 39	1 23 W
1880	57  9	0 21 W	1885	55  3	0 36 W

§ 13. It is often convenient to obtain a formula to express the mean annual change of an element during a given period throughout an area of some size. The usual method is to assume that the change at a place whose latitude is l and longitude λ is given by an expression of the type c + a(l − l₀) + b(λ − λ₀), where a, b, c are constants, l₀ and λ₀, denoting some fixed latitude and longitude which it is convenient to take as point of departure. Supposing observational data available from a series of stations throughout the area, a, b and c can be determined by least squares. As an example, we may take the following slightly modified formula given by Ad. Schmidt^[11] as applicable to Northern Europe for the period 1890 to 1900. ΔD, ΔI and ΔH represent the mean annual changes during this period in westerly declination, in inclination and in horizontal force:—

	′	′	′
ΔD =	−5.24	− 0.071 (l − 50)	+ 0.033 (λ − 10),
ΔI =	−1.58	+ 0.010 (l − 50)	+ 0.036 (λ − 10),
ΔH =	+23.5	− 0.59  (l − 50)	− 0.35  (λ − 10).

Longitude λ is here counted positive to the east. The central position assumed here (lat. 50°, long. 10° E.) falls in the north of Bavaria. In the case of the horizontal force unity represents 1γ. Schmidt found the above formulae to give results in very close agreement with the data at the eight stations which he had employed in determining the constants. These stations ranged from Pavlovsk to Perpignan, and from Stonyhurst to Ekaterinburg in Siberia. Formulae involving the second as well as the first powers of l − l₀ and λ − λ₀ have also been used, e.g., by A. Tanakadate in the Magnetic Survey of Japan.

Table V.—Declination at St Helena and Cape of Good Hope.

St Helena.		Cape of Good Hope.
Date.	Declination.	Date.	Declination.
	°  ′		°  ′
1610	7 13 E	1605	0 30 E
1677	0 40	1609	0 12 W
1691	1  0 W	1675	8 14
1724	7 30	1691	11  0
1775	12 18	1775	21 14
1789	15 30	1792	24 31
1796	15 48	1818	26 31
1806	17 18	1839	29  9
1839	22 17	1842	29  6
1840	22 53	1846	29  9
1846	23 11	1850	29 19
1890	23 57	1857	29 34
		1874	30  4
		1890	29 32
		1903	28 44

Table VI.—Secular Change of Declination in the United States (+ to the West).

Place.	Epoch	1760	70	80	90	1800	10	20	30	40	50	60	70	80	90	1900	50
		′	′	′	′	′	′	′	′	′	′	′	′	′	′	′	′
Eastport, Maine		−1.2	0.0	+1.2	+2.1	+3.2	+4.0	+4.5	+4.9	+5.0	+5.6	+4.5	+3.0	+2.1	+1.0	+1.8	+2.4
Boston, Mass.		−2.7	−1.9	−1.0	0.0	+1.1	+1.9	+2.7	+3.5	+4.2	+4.4	+4.0	+3.3	+3.1	+3.0	+3.2	+3.4
Albany, New York		−4.2	−3.6	−2.7	−1.6	−0.6	+0.6	+1.6	+2.7	+3.6	+4.6	+4.6	+3.9	+4.7	+2.3	+3.4	+3.6
Philadelphia, Penn.		−4.6	−4.2	−3.5	−2.3	−1.3	+0.1	+1.3	+2.5	+3.4	+4.3	+4.2	+4.6	+4.4	+3.4	+3.5	+3.4
Baltimore, Maryland		−3.9	−3.4	−2.7	−2.0	−0.9	0.0	+0.9	+2.0	+2.7	+3.4	+3.9	+4.0	+3.9	+3.6	+3.5	+3.2
Richmond, Virginia		−3.6	−3.2	−2.5	−1.8	−0.9	0.0	+0.9	+1.8	+2.5	+3.1	+3.6	+3.9	+3.8	+3.7	+3.4	+3.2
Columbia, S. Carolina		−3.7	−3.4	−2.9	−2.2	−1.3	−0.5	+0.5	+1.3	+2.2	+2.9	+3.4	+3.8	+3.8	+3.8	+3.6	+1.8
Macon, Georgia		−3.7	−3.6	−3.2	−2.5	−1.8	−0.9	0.0	+0.9	+1.8	+2.5	+3.2	+3.6	+3.9	+3.5	+3.1	+1.2
Tampa, Florida		−3.0	−2.5	−2.0	−1.1	−0.4	+0.4	+1.1	+2.0	+2.5	+3.0	+3.2	+3.5	+3.7	+2.8	+2.9	+1.6
Marquette, Michigan									0.0	+1.4	+2.6	+3.7	+4.7	+5.1	+4.9	+3.8	+2.4
Columbus, Ohio							−0.9	0.0	+0.9	+2.0	+2.9	+3.4	+3.6	+3.7	+3.9	+4.0	+2.4
Bloomington, Illinois							−2.4	−1.5	−0.4	+0.4	+1.5	+2.4	+2.8	+4.2	+3.9	+2.9	+1.0
Lexington, Kentucky							−0.9	0.0	+0.9	+1.8	+2.5	+3.2	+3.6	+3.8	+3.8	+3.4	+1.8
Chattanooga, Tennessee							−0.9	0.0	+0.9	+1.8	+2.5	+3.2	+3.6	+4.0	+3.5	+3.1	+1.6
Little Rock, Arkansas							−2.3	−1.5	−0.9	+0.1	+0.8	+1.7	+2.0	+3.6	+3.7	+2.3	−1.2
Montgomery, Alabama		−3.6	−3.5	−3.1	−2.8	−2.2	−1.5	−0.8	+0.1	+0.8	+1.6	+2.2	+2.8	+3.8	+3.9	+2.6	+0.2
Alexandria, Louisiana							−2.1	−1.6	−0.8	+0.1	+0.8	+1.6	+2.2	+3.6	+3.3	+2.0	−1.4
Northome, Minnesota									−1.7	−0.6	+0.6	+1.7	+2.8	+4.2	+4.4	+3.5	0.0
Jamestown, N. Dakota												+1.0	+1.9	+3.1	+4.8	+1.9	−2.2
Des Moines, Iowa									−1.5	−0.6	+0.6	+1.5	+2.5	+3.8	+4.5	+2.7	−0.6
Douglas, Wyoming												−0.8	0.0	+1.2	+2.3	+0.5	−1.6
Emporia, Kansas												+0.6	+1.6	+2.7	+3.8	+1.7	−1.8
Pueblo, Colorado												−0.3	+0.4	+1.5	+3.1	+0.7	−2.2
Okmulgee, Oklahoma												+0.9	+1.5	+2.7	+3.9	+1.4	−2.4
Santa Rosa, New Mexico												−0.4	+0.4	+1.4	+2.6	+0.4	−2.4
San Antonio, Texas										−1.1	−0.5	−0.5	+1.1	+1.8	+2.7	+0.9	−2.4
Seattle, Washington					−3.3	−3.5	−3.7	−3.7	−3.5	−3.3	−3.0	−2.6	−2.1	−1.3	−1.9	−2.0	−3.2
Wilson Creek, Washington												−2.1	−1.5	−0.4	−1.0	−1.6	−3.2
Detroit, Oregon							−3.8	−3.9	−3.9	−3.7	−3.4	−2.9	−2.5	−1.8	−0.8	−1.8	−3.8
Salt Lake, Utah												−1.1	−0.4	+1.0	+1.0	−0.8	−2.8
Prescott, Arizona												−1.4	−0.7	+0.4	+0.4	−1.2	−3.2
San José, California					−2.6	−2.9	−2.9	−2.9	−2.7	−2.5	−2.3	−2.0	−1.5	−0.8	−0.4	−1.9	−3.8
Los Angeles,   ”					−3.4	−3.4	−3.5	−3.2	−3.0	−2.7	−2.1	−1.6	−1.1	−0.9	−0.3	−1.6	−3.6

Formulae are also wanted to show how the value of an element, or the rate of change of an element, at a particular place has varied throughout a long period. For comparatively short periods it is best to use formulae of the type E = a + bt+ ct ², where E denotes the value of an element t years subsequent to some convenient epoch; a, b, c are constants to be determined from the observational data. For longer periods formulae of the type E = a + b sin (mt + n), where a, b, m and n are constants, have been used by Schott^[12] and others with considerable success. The following examples, due to G. W. Littlehales,^[13] for the Cape of Good Hope, will suffice for illustration:

Declination (West)	= 14°.63 + 15°.00 sin {0.61 (t − 1850) + 77°.8}.
Inclination (South)	= 49°.11 +  8°.75 sin {0.8  (t − 1850) + 34°.3}.

Here t denotes the date. It is perhaps hardly necessary to point out that the extension of any of these empirical formulae—whether to places outside the surveyed area, or to times not included in the period of observation—is fraught with danger, which increases rapidly the further the extrapolation is pushed.

Table VII.—Inclination (northerly) and Horizontal Force at London.

Date.	I.	Date.	I.	Date.	I.	H.	Date.	I.	H.
	°  ′		°  ′		°  ′			°  ′
1576	71 50	1801	70 36.0	1857	68 24.9	.17474	1891	67 33.2	.18193
1600	72  0	1821	70  3.4	1860	69 19.8	.17550	1895	67 25.4	.18278
1676	73 30	1830	69 38.0	1865	68  8.7	.17662	1900	67 11.8	.18428
1723	74 42	1838	69 17.3	1870	67 58.6	.17791	1905	67  3.8	.18510
1773	72 19	1854	68 31.1	1874	67 50.0	.17903	1908	67  0.9	.18515
1786	72  9

Fig. 5.

Bauer has employed a convenient graphical method of illustrating secular change. Radii are drawn from the centre of a sphere parallel to the direction of the freely dipping needle, and are produced to intersect the tangent plane drawn at the point which answers to the mean position of the needle during the epoch under consideration. The curve formed by the points of intersection shows the character of the secular change. Fig. 5 (slightly modified from Nature, vol. 57, p. 181) applies to London. The curve is being described in the clockwise direction. This, according to Bauer’s^[14] own investigation, is the normal mode of description. Schott and Littlehales have found, however, a considerable number of cases where it is difficult to say whether the motion is clockwise or not, while in some stations on both the east and west shores of the Pacific it was clearly anti-clockwise. Fritsche^[15] dealing with the secular changes from 1600 to 1885—as given by his calculated values of the magnetic elements—at 204 points of intersection of equidistant lines of latitude and longitude, found only sixty-three cases in which the motion was unmistakably clockwise, while in twenty-one cases it was clearly the opposite.

§ 14. All the magnetic elements at any ordinary station show a regular variation in the solar day. To separate this from the irregular changes, means of the hourly readings must be formed making use of a number of days. The amplitude of Diurnal Variations. the diurnal change usually varies considerably with the season of the year. Thus a diurnal inequality derived from all the days of the year combined, or from a smaller number of days selected equally from all the months of the year, can give only the average effect throughout the year. Also unless the hours of maxima and minima at a given station are but slightly variable with the season, the result obtained by combining data from all the months of the year may be a hybrid which does not very closely resemble the phenomena in the majority of individual months. This remark applies in particular to the declination at places within the tropics. One consequence is obviously to make the range of a diurnal inequality which answers to the year as a whole less than the arithmetic mean of the twelve ranges obtained for the constituent months. At stations in temperate latitudes, whilst minor differences of type do exist between the diurnal inequalities for different months of the year, the difference is mainly one of amplitude, and the mean diurnal inequality from all the months of the year gives a very fair idea of the nature of the phenomena in any individual month.

Table VIII.—Diurnal Inequality of Declination, mean from whole year (+ to West).

Station.	Jan Mayen.		St Petersburg  and Pavlovsk.		Greenwich.	Kew.		Parc St Maur.	Tiflis.	Kolaba.	Batavia.	Mauritius.	South Vic- toria Land.
Latitude.    Longitude.	71° 0′ N. 8° 28′ W.		59° 41′ N. 30° 29′ E.		51° 28′ N.  0°  0′.	51° 28′ N.  0° 19′ W.		48° 49′ N.  2° 29′ E.	41° 43′ N. 44° 48′ E.	18° 54′ N. 72° 49′ E.	6° 11′ S. 106° 49′ E.	20°  6′ S. 57° 33′ E.	77° 51′ S. 166° 45′ E.
Period.	1882–1883.		1873–1885.		1890–1900.	1890–1900.		1883–1897.	1888–1898.	1894–1901.	1883–1894.	1876–1890.	1902–1903.
	a.	q.	a.	q.	a.	a.	q.	a.	a.	q.	a.	a.	a.	q.
Hour.	′	′	′	′	′	′	′	′	′	′	′	′	′	′
1	− 6.6	−4.2	−1.3	−0.7	−1.4	−1.5	−0.9	−1.4	−0.7	−0.2	+0.1	+0.1	+ 2.0	+ 0.9
2	−10.5	−6.4	−1.2	−0.8	−1.3	−1.4	−0.9	−1.2	−0.6	−0.1	−0.1	+0.1	− 2.1	− 1.8
3	−15.2	−7.8	−1.2	−1.0	−1.3	−1.5	−1.0	−1.2	−0.6	−0.1	−0.1	+0.1	− 5.2	− 4.5
4	−16.9	−8.4	−1.4	−1.3	−1.4	−1.7	−1.3	−1.2	−0.5	−0.1	0.0	+0.2	− 9.4	− 6.8
5	−17.0	−8.1	−1.7	−1.8	−1.7	−2.1	−1.8	−1.6	−0.7	−0.1	0.0	+0.3	−12.2	− 9.0
6	−13.7	−7.0	−1.9	−2.3	−2.1	−2.4	−2.3	−1.9	−1.2	−0.6	+0.1	+0.4	−15.3	−11.7
7	− 9.3	−5.1	−2.2	−2.8	−2.4	−2.7	−2.8	−2.4	−1.9	−1.0	+0.5	+0.6	−17.2	−15.0
8	− 6.8	−3.2	−2.5	−3.2	−2.5	−2.8	−3.1	−2.7	−2.4	−1.2	+1.3	+1.1	−21.5	−17.3
9	− 3.7	−0.6	−2.3	−3.0	−1.9	−2.1	−2.5	−2.3	−2.3	−0.7	+1.7	+1.8	−23.5	−18.1
10	− 2.4	+2.1	−1.0	−1.7	−0.2	−0.3	−0.7	−0.5	−0.9	0.0	+1.5	+1.9	−21.2	−15.8
11	− 0.5	+4.6	+1.0	+0.4	+2.1	+2.2	+1.7	+2.0	+1.0	+0.9	+0.9	+1.3	−15.3	− 9.2
Noon	+ 2.5	+6.5	+3.1	+2.7	+4.2	+4.3	+3.9	+4.2	+2.6	+1.4	+0.1	0.0	− 9.8	− 4.9
1	+ 3.7	+7.3	+4.6	+4.3	+5.1	+5.3	+4.8	+5.3	+3.3	+1.2	−0.6	−1.1	− 3.2	− 0.1
2	+ 6.4	+7.1	+4.9	+4.5	+4.7	+4.9	+4.4	+4.9	+3.1	+0.6	−1.1	−2.0	+ 3.8	+ 5.9
3	+ 7.4	+5.9	+4.1	+3.6	+3.6	+3.7	+3.1	+3.7	+2.3	+0.1	−1.3	−2.3	+11.1	+ 9.5
4	+ 8.5	+4.3	+2.7	+2.3	+2.2	+2.4	+1.8	+2.3	+1.3	−0.2	−1.2	−1.8	+16.6	+12.9
5	+10.6	+3.0	+1.5	+1.3	+1.1	+1.2	+0.7	+1.1	+0.6	−0.1	−0.9	−0.9	+19.9	+14.6
6	+14.2	+2.3	+0.6	+0.7	+0.3	+0.4	+0.2	+0.2	+0.2	0.0	−0.6	−0.1	+22.0	+15.5
7	+15.2	+2.2	0.0	+0.4	−0.3	−0.2	−0.1	−0.4	+0.1	+0.1	−0.4	+0.1	+22.0	+15.9
8	+15.8	+2.6	−0.4	+0.2	−0.9	−0.6	−0.3	−0.9	−0.1	+0.2	−0.2	+0.1	+19.9	+14.6
9	+13.2	+2.6	−1.0	0.0	−1.2	−1.0	−0.5	−1.3	−0.4	+0.1	0.0	+0.1	+16.0	+10.6
10	+ 7.4	+2.0	−1.4	−0.2	−1.5	−1.3	−0.7	−1.5	−0.6	0.0	+0.1	+0.1	+11.6	+ 7.2
11	+ 1.1	+0.5	−1.6	−0.4	−1.6	−1.4	−0.8	−1.6	−0.7	0.0	+0.1	+0.1	+ 7.6	+ 4.2
12	− 3.6	−1.8	−1.5	−0.6	−1.6	−1.5	−0.9	−1.6	−0.8	−0.1	+0.1	+0.1	+ 3.3	+ 1.9
Range	32.8	15.7	7.4	7.7	7.6	8.1	7.9	8.0	5.7	2.6	3.0	4.2	45.5	34.0

Tables VIII. to XI. give mean diurnal inequalities derived from all the months of the year combined, the figures representing the algebraic excess of the hourly value over the mean for the twenty-four hours. The + sign denotes in Table VIII. that the north end of the needle is to the west of its mean position for the day; in Tables IX. to XI. it denotes that the element—the dip being the north or south as indicated—is numerically in excess of the twenty-four hour mean. The letter “a” denotes that all days have been included except, as a rule, those characterized by specially large disturbances. The letter “q” denotes that the results are derived from a limited number of days selected as being specially quiet, i.e. free from disturbance. In all cases the aperiodic or non-cyclic element—indicated by a difference between the values found for the first and second midnights of the day—has been eliminated in the usual way, i.e. by treating it as accumulating at a uniform rate throughout the twenty-four hours. The years from which the data were derived are indicated. The algebraically greatest and least of the hourly values are printed in heavy type; the range thence derived is given at the foot of the tables.

Table IX.—Diurnal Inequality of Horizontal Force, mean from whole year (Unit 1γ = .00001 C.G.S.)

Station.	Jan Mayen.		St Petersburg and Pavlovsk.		Greenwich.	Kew.	Parc St Maur.	Tiflis.	Kolaba.	Batavia.	Mauritius.	S. Victoria Land.
Period.	1882–1883.		1873–1885.		1890–1900.	1890–1900.	1883–1897.	1888–1898.	1894–1901.	1883–1894.	1883–1890.	1902–1903.
	a.	q.	a.	q.	a.	q.	a.	a.	q.	a.	a.	a.
Hour.
1	−57	−22	+ 4	+ 5	+ 4	+ 4	+ 5	+ 3	−10	−11	− 3	−12
2	−64	−24	+ 4	+ 4	+ 3	+ 4	+ 5	+ 3	− 9	−10	− 1	−13
3	−74	−25	+ 4	+ 4	+ 3	+ 4	+ 5	+ 3	− 9	− 8	+ 1	−14
4	−69	−24	+ 4	+ 4	+ 3	+ 4	+ 5	+ 4	− 9	− 7	+ 2	−15
5	−60	−22	+ 5	+ 4	+ 3	+ 4	+ 6	+ 4	− 9	− 5	+ 3	−15
6	−37	−19	+ 4	+ 4	+ 1	+ 2	+ 4	+ 4	− 7	− 1	+ 4	−12
7	−15	−15	+ 2	+ 2	− 3	− 1	+ 1	+ 2	− 1	+ 5	+ 7	− 9
8	− 1	−13	− 3	− 4	− 9	− 7	− 5	− 3	+ 8	+14	+ 9	− 7
9	+ 8	−12	−10	−10	−16	−13	−12	− 8	−19	+24	+ 9	− 3
10	+17	−12	−16	−16	−20	−18	−17	−10	+26	+31	+ 9	+ 3
11	+32	−10	−19	−20	−19	−18	−16	− 7	+30	+35	+ 9	+ 7
Noon	+49	− 4	−17	−18	−13	−12	−12	− 1	+26	+31	+ 8	+12
1	+65	+ 8	−12	−13	− 7	− 7	− 7	+ 4	+19	+22	+ 7	+18
2	+78	+22	− 6	− 6	− 1	− 2	− 4	+ 5	+10	+10	+ 2	+20
3	+89	+37	0	0	+ 2	+ 1	− 1	+ 3	+ 2	− 1	− 2	+19
4	+83	+43	+ 3	+ 3	+ 5	+ 3	0	− 1	− 3	− 9	− 6	+18
5	+68	+49	+ 5	+ 5	+ 7	+ 5	+ 2	− 4	− 7	−13	− 7	+15
6	+37	+43	+ 6	+ 6	+ 9	+ 7	+ 4	− 6	− 8	−14	− 7	+11
7	+13	+30	+ 7	+ 7	+10	+ 8	+ 6	− 4	− 9	−15	− 7	+ 5
8	−11	+15	+ 8	+ 8	+10	+ 8	+ 7	− 1	−10	−16	− 8	+ 0
9	−33	+ 1	+ 9	+ 9	+ 8	+ 7	+ 7	+ 1	−11	−16	− 8	− 4
10	−36	−10	+ 8	+ 9	+ 7	+ 6	+ 6	+ 2	−11	−16	− 8	− 7
11	−40	−16	+ 7	+ 8	+ 6	+ 6	+ 6	+ 3	−10	−15	− 7	− 9
12	−51	−20	+ 6	+ 6	+ 5	+ 5	+ 6	+ 3	−10	−13	− 5	−11
Range	163	74	28	29	30	26	24	15	41	51	17	35

Table X.—Diurnal Inequality of Vertical Force, mean from whole year (Unit 1γ).

Station.	Jan Mayen.		St Petersburg and Pavlovsk.		Greenwich.	Kew.	Parc St Maur.	Tiflis.	Kolaba.	Batavia.	Mauritius.	South Vic- toria Land.
Period.	1882–1883.		1873–1885.		1890–1900.	1891–1900.	1883–1897.	1888–1898.	1894–1901.	1883–1894.	1884–1890.	1902–1903.
	a.	q.	a.	q.	a.	q.	a.	a.	q.	a.	a.	a.
Hour
1	+65	+ 3	− 7	− 1	− 3	+ 1	0	+ 2	+ 4	+ 7	+ 2	+13
2	+65	+ 2	− 7	− 1	− 4	+ 1	0	+ 2	+ 4	+ 5	+ 2	+12
3	+56	− 1	− 7	− 1	− 4	0	− 1	+ 1	+ 3	+ 4	+ 2	+10
4	+37	− 5	− 6	0	− 3	0	0	+ 1	+ 3	+ 3	+ 2	+ 8
5	+16	− 7	− 5	0	− 2	+ 1	0	+ 2	+ 5	+ 2	+ 2	+ 3
6	− 7	− 8	− 4	0	− 1	+ 1	+ 1	+ 3	+ 7	+ 1	+ 2	0
7	−17	− 6	− 3	0	0	0	+ 1	+ 3	+ 6	0	+ 3	0
8	−14	− 4	− 2	0	0	− 1	0	+ 3	0	− 3	+ 4	− 2
9	− 9	0	− 3	− 1	− 3	− 4	− 4	− 1	− 8	−11	+ 5	− 6
10	− 6	+ 5	− 2	− 2	− 6	− 8	− 8	− 7	−14	−20	+ 3	−13
11	− 6	+10	− 3	− 4	− 9	−11	−12	−11	−15	−26	0	−17
Noon	−10	+16	− 3	− 5	−10	−11	−12	−11	−10	−27	− 4	−20
1	−13	+21	− 1	− 4	− 6	− 8	− 9	− 9	− 3	−21	− 7	−20
2	−24	+23	+ 2	− 1	0	− 3	− 3	− 5	+ 1	−13	− 9	−16
3	−31	+20	+ 8	+ 2	+ 5	+ 2	+ 2	− 1	+ 4	− 4	− 8	−12
4	−40	+13	+ 9	+ 3	+ 8	+ 5	+ 6	+ 1	+ 3	+ 4	− 5	− 6
5	−48	+ 2	+10	+ 3	+ 9	+ 6	+ 7	+ 3	0	+10	− 3	− 1
6	−53	− 9	+10	+ 3	+10	+ 7	+ 8	+ 4	0	+13	0	+ 3
7	−47	−18	+ 9	+ 3	+ 9	+ 6	+ 7	+ 3	0	+14	0	+ 6
8	−36	−20	+ 8	+ 3	+ 7	+ 5	+ 6	+ 3	+ 1	+14	+ 1	+ 9
9	− 7	−19	+ 6	+ 2	+ 5	+ 5	+ 5	+ 3	+ 2	+14	+ 2	+11
10	+18	−13	+ 3	+ 2	+ 3	+ 4	+ 3	+ 3	+ 3	+13	+ 2	+12
11	+42	− 5	− 2	0	0	+ 3	+ 2	+ 3	+ 3	+11	+ 2	+12
12	+54	0	− 5	− 1	− 2	+ 2	+ 1	+ 2	+ 3	+ 9	+ 2	+13
Range	118	43	17	8	20	18	20	15	22	41	14	33

When comparing results from different stations, it must be remembered that the disturbing forces required to cause a change of 1′ in declination and in dip vary directly, the former as the horizontal force, the latter as the total force. Near a magnetic pole the horizontal force is relatively very small, and this accounts, at least partly, for the difference between the declination phenomena at Jan Mayen and South Victoria Land on the one hand and at Kolaba, Batavia and Mauritius on the other. There is, however, another cause, already alluded to, viz. the variability in the type of the diurnal inequality in tropical stations. With a view to illustrating this point Table XII. gives diurnal inequalities of declination for June and December for a number of stations lying between 45° N. and 45° S. latitude. Some of the results are represented graphically in fig. 6, plus ordinates representing westerly deflection. At the northmost station, Toronto, the difference between the two months is mainly a matter of amplitude, the range being much larger at midsummer than at midwinter. The conspicuous phenomenon at both seasons is the rapid swing to the west from 8 or 9 a.m. to 1 or 2 p.m. At the extreme southern station, Hobart—at nearly equal latitude—the rapid diurnal movement is to the east, and so in the opposite direction to that in the northern hemisphere, but it again takes place at nearly the same hours in June (midwinter) as in December. If, however, we take a tropical station such as Trivandrum or Kolaba, the phenomena in June and December are widely different in type. At Trivandrum—situated near the magnetic equator in India—we have in June the conspicuous forenoon swing to the west seen at Toronto, occurring it is true slightly earlier in the day; but in December at the corresponding hours the needle is actually swinging to the east, just as it is doing at Hobart. In June the diurnal inequality of declination at tropical stations—whether to the north of the equator like Trivandrum, or to the south of it like Batavia—is on the whole of the general type characteristic of temperate regions in the northern hemisphere; whereas in December the inequality at these stations resembles that of temperate regions in the southern hemisphere. Comparing the inequalities for June in Table XII. amongst themselves, and those for December amongst themselves, one can trace a gradual transformation from the phenomena seen at Toronto to those seen at Hobart. At a tropical station the change from the June to the December type is probably in all cases more or less gradual, but at some stations the transition seems pretty rapid.

Table XI.—Diurnal Inequality of Inclination mean from whole year.

Station.	Jan Mayen.		St Petersburg and Pavlovsk.		Greenwich.	Kew.	Parc St Maur.	Tiflis.	Kolaba.	Batavia.	Mauritius.	South Vic- toria Land.
End Dipping	North.		North.		North.	North.	North.	North.	North.	South.	South.	South.
Period.	1882–1883.		1873–1885.		1890–1900.	1891–1900.	1883–1897.	1888–1898.	1894–1901.	1883–1894.	1884–1890.	1902–1903.
	a.	q.	a.	q.	a.	q.	a.	a.	q.	a.	a.	a.
Hour	′	′	′	′	′	′	′	′	′	′	′	′
1	+4.6	+1.5	−0.5	−0.3	−0.4	−0.3	−0.3	−0.1	+0.6	+0.9	+0.3	+0.6
2	+5.0	+1.6	−0.5	−0.3	−0.3	−0.2	−0.3	−0.1	+0.6	+0.8	+0.2	+0.7
3	+5.6	+1.6	−0.5	−0.3	−0.3	−0.2	−0.3	−0.1	+0.5	+0.6	0.0	+0.7
4	+5.0	+1.5	−0.4	−0.3	−0.3	−0.2	−0.4	−0.2	+0.5	+0.5	−0.0	+0.7
5	+4.2	+1.4	−0.5	−0.3	−0.2	−0.2	−0.4	−0.2	+0.7	+0.3	−0.1	+0.7
6	+2.4	+1.2	−0.4	−0.3	−0.1	−0.1	−0.3	−0.1	+0.8	+0.1	−0.2	+0.5
7	+0.7	+0.9	−0.2	−0.1	+0.2	+0.1	0.0	0.0	+0.5	−0.2	−0.3	+0.4
8	−0.1	+0.8	+0.1	+0.3	+0.6	+0.4	+0.4	+0.3	−0.2	−0.8	−0.4	+0.3
9	−0.7	+0.8	+0.6	+0.6	+1.0	+0.8	+0.7	+0.5	−1.2	−1.7	−0.4	+0.1
10	−1.2	+0.9	+1.0	+1.0	+1.1	+1.0	+0.9	+0.3	−1.9	−2.7	−0.5	−0.2
11	−2.2	+0.8	+1.2	+1.2	+1.0	+0.9	+0.7	0.0	−2.1	−3.3	−0.6	−0.4
Noon	−3.4	+0.4	+1.1	+1.1	+0.6	+0.6	+0.4	−0.5	−1.6	−3.1	−0.7	−0.7
1	−4.5	−0.2	+0.7	+0.7	+0.3	+0.2	+0.2	−0.6	−0.8	−2.4	−0.8	−0.9
2	−5.6	−1.2	+0.4	+0.4	+0.1	+0.1	+0.2	−0.5	−0.2	−1.3	−0.6	−1.0
3	−6.3	−2.2	+0.2	+0.1	0.0	0.0	+0.2	−0.3	+0.3	−0.2	−0.3	−1.0
4	−6.1	−2.9	0.0	−0.1	−0.1	−0.1	+0.2	+0.1	+0.3	+0.7	+0.1	−0.9
5	−5.1	−3.2	−0.1	−0.3	−0.2	−0.2	+0.1	+0.4	+0.2	+1.3	+0.4	−0.7
6	−3.1	−2.9	−0.2	−0.3	−0.3	−0.3	0.0	+0.5	+0.2	+1.5	+0.5	−0.5
7	−1.7	−2.2	−0.3	−0.4	−0.4	−0.4	−0.2	+0.4	+0.3	+1.6	+0.5	−0.2
8	+0.3	−1.3	−0.3	−0.5	−0.4	−0.4	−0.3	+0.2	+0.4	+1.6	+0.6	0.0
9	+2.0	−0.3	−0.4	−0.6	−0.4	−0.4	−0.3	+0.1	+0.5	+1.6	+0.6	+0.2
10	+2.5	+0.5	−0.5	−0.6	−0.4	−0.3	−0.3	0.0	+0.6	+1.5	+0.6	+0.4
11	+3.0	+1.0	−0.5	−0.6	−0.4	−0.3	−0.3	0.0	+0.6	+1.4	+0.5	+0.5
12	+4.0	+1.3	−0.5	−0.4	−0.4	−0.3	−0.3	−0.1	+0.6	+1.2	+0.4	+0.6
Range	11.9	4.8	1.7	1.8	1.5	1.4	1.3	1.1	2.9	4.9	1.4	1.7

Table XII.—Diurnal Inequality of Declination (+ to West).

Station.	Toronto.		Kolaba.		Trivandrum.		Batavia.		St Helena.		Mauritius.		Cape.		Hobart.
Month.	June.	Dec.	June.	Dec.	June.	Dec.	June.	Dec.	June.	Dec.	June.	Dec.	June.	Dec.	June.	Dec.
Hour	′	′	′	′	′	′	′	′	′	′	′	′	′	′	′	′
1	−0.4	−0.1	−0.3	0.0	−0.3	−0.1	+0.1	+0.1	−0.1	−0.4	0.0	+0.1	−0.4	−0.7	+0.8	+1.1
2	−0.2	+0.4	−0.3	+0.1	−0.4	+0.1	−0.1	+0.1	−0.2	−0.1	−0.2	+0.2	−0.5	−0.4	+0.3	+1.1
3	−0.2	−0.1	−0.3	+0.1	−0.4	+0.3	−0.2	+0.2	−0.2	+0.1	−0.2	+0.4	−0.7	−0.1	−0.1	+1.0
4	−1.2	−0.4	−0.3	+0.3	−0.5	+0.5	−0.3	+0.3	−0.3	+0.3	−0.2	+0.7	−0.6	+0.3	−0.1	+1.1
5	−2.9	−0.6	−0.7	+0.4	−0.7	+0.7	−0.3	+0.5	−0.5	+0.6	−0.3	+1.0	−0.7	+1.0	0.0	+1.7
6	−5.2	−0.6	−1.6	+0.5	−1.6	+1.1	−0.5	+1.2	−1.0	+0.9	−0.4	+1.7	−1.0	+2.2	0.0	+2.7
7	−6.2	−0.9	−2.2	+0.7	−1.7	+1.4	−1.1	+2.0	−2.2	+1.9	−1.1	+2.6	−1.6	+3.3	−0.1	+4.4
8	−6.0	−1.2	−2.1	+0.2	−1.1	+0.9	−0.4	+2.3	−1.5	+2.2	−1.0	+2.4	−0.8	+3.6	+0.1	+5.6
9	−4.4	−1.8	−1.1	−0.1	−0.2	+0.5	+0.5	+2.0	−0.3	+1.3	+0.2	+2.0	+0.7	+3.1	+0.6	+5.6
10	−1.5	−1.1	0.0	−0.2	+0.6	+0.3	+0.9	+1.3	+0.3	+0.2	+1.2	+1.1	+1.6	+1.6	+1.2	+3.6
11	+2.1	+0.6	+1.2	0.0	+1.2	+0.1	+1.0	+0.4	+0.5	−1.0	+1.4	0.0	+1.5	+0.1	+1.0	+0.7
Noon	+4.8	+2.2	+2.1	0.0	+1.4	−0.4	+0.7	−0.6	+0.3	−1.4	+1.0	−1.4	+0.8	−1.0	−0.1	−2.6
1	+6.1	+3.2	+2.0	−0.2	+1.1	−0.8	+0.3	−1.4	+0.3	−1.2	+0.1	−2.2	+0.3	−1.8	−1.4	−5.1
2	+6.1	+3.2	+1.6	−0.3	+0.7	−0.9	−0.2	−1.8	+0.2	−0.4	−0.9	−2.5	−0.3	−1.9	−2.2	−6.2
3	+5.2	+2.4	+0.9	−0.3	+0.3	−0.9	−0.7	−1.9	+0.2	+0.4	−1.5	−2.2	−0.3	−1.4	−2.4	−5.8
4	+3.6	+1.5	+0.2	−0.3	+0.1	−0.8	−0.8	−1.6	+0.7	+0.6	−1.3	−1.6	+0.2	−0.8	−1.6	−4.8
5	+1.8	+0.5	0.0	−0.2	0.0	−0.4	−0.5	−1.2	+1.1	+0.4	−0.3	−1.0	+0.5	−0.8	−0.7	−3.3
6	+0.7	−0.1	+0.1	−0.2	+0.2	−0.4	−0.1	−0.7	+1.0	+0.1	+0.5	−0.5	+0.5	−0.6	−0.4	−1.9
7	0.0	−0.8	+0.3	−0.2	+0.5	−0.4	+0.1	−0.6	+0.6	−0.4	+0.7	−0.3	+0.4	−0.8	0.0	−1.0
8	0.0	−1.2	+0.4	−0.1	+0.5	−0.3	+0.2	−0.5	+0.5	−0.7	+0.7	−0.3	+0.3	−0.9	+0.5	−0.3
9	−0.5	−1.4	+0.3	−0.1	+0.4	−0.2	+0.4	−0.3	+0.4	−0.9	+0.6	−0.2	+0.2	−0.9	+1.1	0.0
10	−0.5	−1.7	+0.1	0.0	+0.2	−0.1	+0.4	−0.1	+0.2	−1.0	+0.4	−0.1	+0.1	−1.0	+1.3	+0.6
11	−0.7	−1.1	−0.1	−0.1	0.0	−0.1	+0.3	0.0	+0.1	−0.8	+0.3	0.0	0.0	−1.0	+1.3	+0.9
12	−0.6	−0.7	−0.2	−0.1	−0.2	−0.1	+0.2	+0.1	−0.1	−0.6	+0.1	+0.1	−0.2	−1.0	+1.1	+1.2
Range	12.3	5.0	4.3	1.0	3.1	2.3	2.1	4.2	3.3	3.6	2.9	5.1	3.2	5.5	3.7	11.8

§ 15. In the case of the horizontal force there are, as Table IX. shows, two markedly different types of diurnal inequality. In the one type, exemplified by Pavlovsk or Greenwich, the force is below its mean value in the middle of the day; it has a principal minimum about 10 or 11 a.m., and morning and evening maxima, the latter usually the largest. In the other type, exemplified by Kolaba or Batavia, the horizontal force is above its mean in the middle of the day, and has a maximum about 11 a.m. The second type may be regarded as the tropical type. At tropical stations, such as Kolaba, Batavia, Manila and St Helena, the type is practically the same in summer as in winter, and is the same whether the station is north or south of the equator. Similarly, what we may call the temperate type is seen—with comparatively slight modifications—both in summer and winter at stations such as Greenwich or Pavlovsk. In winter, it is true, the pronounced daily minimum is a little later and the early morning maximum is relatively more important than in summer. There is not, as in the case of the declination, any essential difference between the phenomena at temperate stations in the northern and southern hemispheres.

Fig. 6.

With diminishing latitude, there is a gradual transition from the temperate to the tropical type of horizontal force diurnal variation, and at stations whose latitude is under 45° there is a very appreciable variation in type with the season. The mean diurnal variation for the year at Tiflis in Table IX. really represents a struggle between the two types, in which on the whole the temperate type prevails. If we take the diurnal variations at Tiflis for midsummer and midwinter, we find the former essentially of the temperate, the latter essentially of the tropical type. A similar conflict may be seen in the mean diurnal inequality for the year at the Cape of Good Hope, but there the tropical type on the whole predominates, and it prevails more at midwinter than at midsummer. Toronto and Hobart, though similar in latitude to Tiflis, show a closer approach to the temperate type. Still at both stations the hours during which the force is below its mean value tend to extend back towards midnight, especially at midsummer. The amplitude of the horizontal force range appears less at intermediate stations, such as Tiflis, than at stations in either higher or lower latitudes. There is a very great difference in this respect between the north and the south of India.

§ 16. In the case of the vertical force in higher temperate latitudes—at Pavlovsk for instance—the diurnal inequalities from “all” and from “quiet” days differ somewhat widely in amplitude and slightly even in type. In mean latitudes, e.g. at Tiflis, there is often a well marked double period in the mean diurnal inequality for the whole year; but even at Tiflis this is hardly, if at all, apparent in the winter months. In the summer months the double period is distinctly seen at Kew and Greenwich, though the evening maximum is always pre-eminent. Speaking generally, the time of the minimum, or principal minimum, varies much less with the season than that of the maximum. At Kew, for instance, on quiet days the minimum falls between 11 a.m. and noon in almost all the months of the year, but the time of the maximum varies from about 4 p.m. in December to 7 p.m. in June. At Kolaba the time of the minimum is nearly independent of the season; but the changes from positive to negative in the forenoon and from negative to positive in the afternoon are some hours later in winter than in summer. At Batavia the diurnal inequality varies very little in type with the season, and there is little evidence of more than one maximum and minimum in the day. At Batavia, as at Kolaba, negative values occur near noon; but it must be remembered that while at Kolaba and more northern stations vertical force urges the north pole of a magnet downwards, the reverse is true of Batavia, as the dip is southerly. At St Helena vertical force is below its mean value in the forenoon, but the change from − to + occurs at noon, or but little later, both in winter and summer. At the Cape of Good Hope the phenomena at midsummer are similar to those at Kolaba, the force being below its mean value from about 9 a.m. to 3 p.m. and above it throughout the rest of the day; but at midwinter there is a conspicuous double period, the force being below its mean from 1 a.m. to 7 a.m. as well as from 11 a.m. to 3 p.m., and thus resembling the all-day annual results at Greenwich. At Hobart vertical force is below its mean value from 1 a.m. to 9 a.m. at midsummer, and from 4 a.m. to noon at midwinter; while the force is above its mean persistently throughout the afternoon both in summer and winter, there is at midwinter a well marked secondary minimum about 6 p.m., almost the same hour as that at which the maximum for the day is observed in summer.

Table XIII.—Range of the Diurnal Inequality of Declination.

Place.	Period.	Jan.	Feb.	March.	April.	May.	June.	July.	Aug.	Sept.	Oct.	Nov.	Dec.
		′	′	′	′	′	′	′	′	′	′	′	′
Pavlovsk	1890–1900 a	4.93	6.15	8.58	10.93	12.18	12.27	11.82	11.38	8.70	6.87	5.54	4.63
”	1890–1900 q	2.96	4.20	8.73	11.28	12.89	13.28	12.31	11.70	9.37	6.91	3.95	2.66
Ekatarinburg	1890–1900 a	3.33	4.32	7.63	11.19	11.82	11.58	11.09	10.45	8.13	5.60	3.73	3.14
Greenwich	1865–1896 a	5.87	7.07	9.40	11.42	10.55	10.90	10.82	10.93	9.66	8.15	6.41	5.15
Kew	1890–1900 a	4.92	6.06	9.08	10.95	10.66	10.92	10.59	11.01	9.49	7.73	5.37	4.46
”	1890–1900 q	4.07	4.76	8.82	10.57	10.92	10.62	10.18	11.01	9.76	7.51	4.75	3.34
Toronto	1842–1848 a	5.96	6.05	9.18	9.94	11.55	12.34	12.21	13.14	10.76	6.96	6.32	4.97
Manila	1890–1900 a	1.79	1.09	2.13	3.02	3.84	3.94	4.21	4.89	4.53	1.83	0.85	1.33
Trivandrum	1853–1864 a	2.06	1.48	0.79	1.67	2.90	3.06	3.06	3.64	3.31	1.27	2.14	2.33
Batavia	1884–1899 a	4.18	4.64	3.57	2.93	2.38	2.03	2.31	3.16	3.80	4.51	4.50	4.19
St Helena	1842–1847 a	3.72	5.19	4.93	3.30	2.64	3.24	3.42	3.59	2.40	4.43	4.05	3.54
Mauritius	1876–1890 a	5.2	6.1	6.3	4.7	4.1	2.9	3.4	4.9	5.0	5.5	5.6	5.1
Cape	1841–1846 a	5.14	8.21	7.27	5.00	3.91	3.21	3.54	4.98	4.33	5.96	6.36	5.47
Hobart	1841–1848 a	11.66	11.80	9.50	7.26	4.56	3.70	4.61	5.89	8.24	11.01	12.05	11.81

§ 17. Variations of inclination are connected with those of horizontal and vertical force by the relation

δI = 1/2 sin 2I {V⁻¹δV − H⁻¹ δH}.

Thus in temperate latitudes where V is considerably in excess of H, whilst diurnal changes in V are usually less than those in H, it is the latter which chiefly dominate the diurnal changes in inclination. When the H influence prevails, I has its highest values at hours when H is least. This explains why the dip is above its mean value near midday at stations in Table XI. from Pavlovsk to Parc St Maur. Near the magnetic equator the vertical force has the greater influence. This alone would tend to make a minimum dip in the late forenoon, and this minimum is accentuated owing to the altered type of the horizontal force diurnal variation, whose maximum now coincides closely with the minimum in the vertical force. This accounts for the prominence of the minimum in the diurnal variation of the inclination at Kolaba and Batavia, and the large amplitude of the range. Tiflis shows an intermediate type of diurnal variation; there is a minimum near noon, as in tropical stations, but inclination is also below its mean for some hours near midnight. The type really varies at Tiflis according to the season of the year. In June—as in the mean equality from the whole year—there is a well marked double period; there is a principal minimum at 2 p.m. and a secondary one about 4 a.m.; a principal maximum about 9 a.m. and a secondary one about 6 p.m. In December, however, only a single period is recognizable, with a minimum about 8 a.m. and a maximum about 7 p.m. The type of diurnal inequality seen at the Cape of Good Hope does not differ much from that seen at Batavia. Only a single period is clearly shown. The maximum occurs about 8 or 9 p.m. throughout the year. The time of the minimum is more variable; at midsummer it occurs about 11 a.m., but at midwinter three or four hours later. At Hobart the type varies considerably with the season. In June (midwinter) a double period is visible. The principal minimum occurs about 8 a.m., as at the Cape. But, corresponding to the evening maximum seen at the Cape, there is now only a secondary maximum, the principal maximum occurring about 1 p.m. At midsummer the principal maximum is found—as at Kew or Greenwich—about 10 or 11 a.m., the principal minimum about 4 p.m.

§ 18. Even at tropical stations a considerable seasonal change is usually seen in the amplitude of the diurnal inequality in at least one of the magnetic elements. At stations in Europe, and generally in temperate latitudes, the amplitude varies notably in all the elements. Table XIII. gives particulars of the inequality range of declination derived from hourly readings at selected stations, arranged in order of latitude from north to south. The letters “a” and “q” are used in the same sense as before. At temperate stations in either hemisphere—e.g. Pavlovsk, Greenwich or Hobart—the range is conspicuously larger in summer than in winter. In northern temperate stations a decided minimum is usually apparent in December. There is, on the other hand, comparatively little variation in the range from April to August. Sometimes, as at Kew and Greenwich, there is at least a suggestion of a secondary minimum at midsummer. Manila and Trivandrum show a transition from the December minimum, characteristic of the northern stations, to the June minimum characteristic of the southern, there being two conspicuous minima in February or March and in November or October. At St Helena there are two similar minima in May and September, while a third apparently exists in December. It will be noticed that at both Pavlovsk and Kew the annual variation in the range is specially prominent in the quiet day results.

Table XIV. gives a smaller number of data analogous to those of Table XIII., comprising inequality ranges for horizontal force, vertical force and inclination. In some cases the number of years from which the data were derived seems hardly sufficient to give a smooth annual variation. It should also be noticed that unless the same group of years is employed the data from two stations are not strictly comparable. The difference between the all and quiet day vertical force data at Pavlovsk is remarkably pronounced. The general tendency in all the elements is to show a reduced range at midwinter; but in some cases there is also a distinct reduction in the range at midsummer. This double annual period is particularly well marked at Batavia.

Table XIV.—Ranges in the Diurnal Inequalities.

			Jan.	Feb.	March.	April.	May.	June.	July.	Aug.	Sept.	Oct.	Nov.	Dec.
H (unit 1γ)
Pavlovsk	1890–1900	a	12	20	32	46	47	49	49	44	39	32	17	11
”	”	q	12	17	31	42	45	45	42	40	37	31	17	10
Ekatarinburg	”	a	11	15	29	37	40	40	39	36	33	27	13	9
Kew	”	q	15	17	26	36	38	39	38	38	35	27	20	11
Toronto	1843–1848	a	23	21	24	28	29	29	26	28	41	25	21	20
Batavia	1883–1898	a	49	47	54	60	51	48	50	53	58	52	43	40
St Helena	1843–1847	a	43	41	48	53	46	40	40	45	41	40	40	32
Mauritius	1883–1890	a	21	15	21	23	20	21	20	22	20	21	21	20
Cape of Good Hope	1841–1846	a	13	10	13	13	15	16	14	18	21	14	17	20
Hobart	1842–1848	a	42	43	34	28	19	17	22	23	23	35	39	42
V (unit 1γ)
Pavlovsk	1890–1900	a	15	27	29	24	26	20	23	19	23	20	18	14
”	”	q	4	5	9	13	13	12	13	10	9	7	5	4
Ekatarinburg	”	a	10	15	17	21	22	19	20	16	14	13	11	9
Kew	1891–1900	q	7	10	20	25	31	27	28	23	20	15	9	6
Toronto	1843–1848	a	12	14	17	23	26	14	27	32	34	25	19	18
Batavia	1883–1898	a	42	48	48	45	31	31	32	29	41	50	40	33
St Helena	1843–1847	a	16	13	12	14	13	11	17	11	17	11	15	18
Mauritius	1884–1890	a	12	16	18	15	14	13	15	21	20	16	13	11
Cape of Good Hope	1841–1846	a	29	47	41	38	21	12	14	19	19	35	33	28
Hobart	1842–1848	a	25	27	22	23	24	21	22	28	26	22	23	27
Inclination			′	′	′	′	′	′	′	′	′	′	′	′
Pavlovsk	1890–1900	a	0.97	1.24	2.07	2.79	2.72	2.88	2.85	2.64	2.52	2.18	1.20	0.89
Ekatarinburg	”	a	0.79	0.94	1.70	2.08	2.25	2.19	2.18	2.08	2.00	1.70	0.88	0.69
Kew	”	q	0.98	1.01	1.38	1.86	2.05	2.02	2.05	2.15	1.98	1.57	1.27	0.63
Toronto	1843–1848	a	1.15	0.94	1.19	1.23	1.31	1.37	1.13	1.26	1.87	1.16	1.09	1.05
Batavia	1883–1898	a	4.88	5.22	5.56	5.62	4.21	4.05	4.24	4.17	5.13	5.58	4.51	3.85
Cape of Good Hope	1842–1846	a	1.55	2.29	2.23	2.23	1.60	1.41	1.54	1.70	1.86	2.03	1.55	2.04
Hobart	1842–1848	a	1.95	2.16	1.72	1.62	1.23	1.16	1.28	1.42	1.39	1.75	2.04	2.10

§ 19. When discussing diurnal inequalities it is sometimes convenient to consider the components of the horizontal force in and perpendicular to the astronomical meridian, rather than the horizontal force and declination. If N and W be the components of H to astronomical north and west, and D the westerly declination, N = H cos D, W = H sin D. Thus corresponding small variations in N, W, H and D are connected by the relations:—

δN = cos DδH − H sin DδD, δW = sin DδH + H cos DδD.

If δH and δD denote the departures of H and D at any hour of the day from their mean values, then δN and δW represent the corresponding departures of N and W from their mean values. In this way diurnal inequalities may be calculated for N and W when those for H and D are known. The formulae suppose δD to be expressed in absolute measure, i.e. 1′ of arc has to be replaced by 0.0002909. If we take as an example a station at which H is .185 then HδD = .0000538 (number of minutes in δD). In other words, employing 1γ as unit of force, one replaces HδD by 5.38δD, where δD represents declination change expressed as usual in minutes of arc. In calculating diurnal inequalities for N and W, one ought, strictly speaking, to assign to H and D the exact mean values belonging to these elements for the month or the year being dealt with. For practical purposes, however, a slight departure from the true mean values is immaterial, and one can make use of a constant value for several successive years without sensible error. As an example, Table XV. gives the mean diurnal inequality for the whole year in N and W at Falmouth, as calculated from the 12 years 1891 to 1902. The unit employed is 1γ.

The data in Table XV. are closely similar to corresponding Kew data, and are presumably fairly applicable to the whole south of England for the epoch considered. At Falmouth there is comparatively little seasonal variation in the type of the diurnal variation in either N or W. The amplitude of the diurnal range varies, however, largely with the season, as will appear from Table XVI., which is based on the same 12 years as Table XV.

Diurnal inequalities in N and W lend themselves readily to the construction of what are known as vector diagrams. These are curves showing the direction and intensity at each hour of the day of the horizontal component of the disturbing force to which the diurnal inequality may be regarded as due. Figs. 7 and 8, taken from the Phil. Trans. vol. 204A, will serve as examples. They refer to the mean diurnal inequalities for the months stated at Kew (1890 to 1900) and Falmouth (1891 to 1902), thick lines relating to Kew, thin to Falmouth. NS and EW represent the geographical north-south and east-west directions; their intersection answers to the origin (thick lines for Kew, thin for Falmouth). The line from the origin to M represents the magnetic meridian. The line from the origin to any cross—the number indicating the corresponding hour counted from midnight as 0—represents the magnitude and direction at that hour of the horizontal component of the disturbing force to which the diurnal inequality may be assigned. The cross marks the point whose rectangular co-ordinates are the values of δN and δW derived from the diurnal inequalities of these elements. In figs. 7 and 8 the distances of the points N, E, S, W from their corresponding origin represents 10γ. The tendency to form a loop near midnight, seen in the November and December curves, is characteristic of the winter months at Kew and Falmouth. The shape is less variable in summer than in winter; but even in summer the portion answering to the hours 6 p.m. to 6 a.m. varies a good deal. The object of presenting the Kew and Falmouth curves side by side is to emphasize the close resemblance between the magnetic phenomena at places in similar latitudes, though over 200 miles apart and exhibiting widely different ranges for their meteorological elements. With considerable change of latitude however the shape of vector diagrams changes largely.

Table XV.—Diurnal Inequalities in N. and W. at Falmouth (unit 1γ).

Hour.

1

2

3

4

5

6

7

8

9

10

11

12

N.${\Big \{}$

a.m.

+ 6

+ 5

+ 5

+ 5

+ 6

+ 6

+ 5

+ 1

− 6

−14

−20

−20

p.m.

−17

−12

− 6

− 1

+ 3

+ 6

+ 9

+ 9

+ 9

+ 8

+ 7

+ 7

W.${\Big \{}$

a.m.

− 2

− 2

− 3

− 4

− 6

− 9

−13

−17

−19

−13

− 3

+11

p.m.

+20

+22

+17

+11

+ 6

+ 4

+ 2

+ 1

0

− 1

− 2

− 2

§ 20. Any diurnal inequality can be analysed into a series of harmonic terms whose periods are 24 hours and submultiples thereof. The series may be expressed in either of the equivalent Fourier Series. forms:—

a1 cos t + b1 sin t + a2 cos 2t + b2 sin 2t + . . .

(i)

c1 sin (t + α1) + c2 sin (2t + α2) + . . . .

(ii)

Table XVI.—Ranges in Diurnal Inequalities at Falmouth (unit 1γ).

	Jan.	Feb.	March.	April.	May.	June.	July.	Aug.	Sept.	Oct.	Nov.	Dec.
N.	21	23	30	39	39	37	37	39	36	32	24	15
W.	20	24	46	54	55	55	54	56	51	39	24	15

In both forms t denotes time, counted usually from midnight, one hour of time being interpreted as 15° of angle. Form (i) is that utilized in actually calculating the constants a, b, . . . Once the a, b, . . . constants are known, the c, α, . . . constants are at once derivable from the formulae:—

tan α_n = a_n / b_n; c_n = a_n / sin α_n = b_n / cos α_n = √(a_n² + b_n²).

The a, b, c, α constants are called sometimes Fourier, sometimes Bessel coefficients.

(From Phil. Trans.)

Fig. 7.

By taking a sufficient number of terms a series can always be obtained which will represent any set of diurnal inequality figures; but unless one can obtain a close approach to the observational figures from the terms possessing the periods 24, 12, 8 and 6 hours the physical significance and general utility of the analysis is somewhat problematical. In the case of the magnetic elements, the 24 and 12 hour terms are usually much the more important; the 24–hour term is generally, but by no means always, the larger of the two. The c constants give the amplitudes of the harmonic terms or waves, the α constants the phase angles. An advance of 1 hour in the time of occurrence of the first (and subsequent, if any) maximum and minimum answers to an increase of 15° in α₁ of 30° in α₂, of 45° in α₃, of 60° in α₄ and so on. In the case of magnetic elements the phase angles not infrequently possess a somewhat large annual variation. It is thus essential for a minute study of the phenomena at any station to carry out the analysis for the different seasons of the year, and preferably for the individual months. If the a and b constants are known for all the individual months of one year, or for all the Januarys of a series of years, we have only to take their arithmetic means to obtain the corresponding constants for the mean diurnal inequality of the year, or for the diurnal inequality of the average January of the series of years. This, however, is obviously not true of the c or α constants, unless the phase angle is absolutely unchanged throughout the contributory months or years. This is a point requiring careful attention, because when giving values of c and α for the whole year some authorities give the arithmetic mean of the c’s and α’s calculated from the diurnal inequalities of the individual months of the year, others give the values obtained for c and α from the mean diurnal inequality of the whole year. The former method inevitably supplies a larger value for c than the latter, supposing α to vary with the season. At some observatories, e.g. Greenwich and Batavia, it has long been customary to publish every year values of the Fourier coefficients for each month, and to include other elements besides the declination. For a thoroughly satisfactory comparison of different stations, it is necessary to have data from one and the same epoch; and preferably that epoch should include at least one 11–year period. There are, however, few stations which can supply the data required for such a comparison and we have to make the best of what is available. Information is naturally most copious for the declination. For this element E. Engelenburg^[16] gives values of C₁, C₂, C₃, C₄, and of α₁, α₂, α₃, α₄ for each month of the year for about 50 stations, ranging from Fort Rae (62° 6′ N. lat.) to Cape Horn (55° 5′ S. lat.). From the results for individual stations, Engelenburg derives a series of means which he regards as representative of 11 different zones of latitude. His data for individual stations refer to different epochs, and some are based on only one year’s observations. The original observations also differ in reliability; thus the results are of somewhat unequal value. The mean results for Engelenburg’s zones must naturally have some of the sources of uncertainty reduced; but then the fundamental idea represented by the arrangement in zones is open to question. The majority of the data in Table XVII. are taken from Engelenburg, but the phase angles have been altered so as to apply to westerly declination. The stations are arranged in order of latitude from north to south; in a few instances results are given for quiet days. The figures represent in all cases arithmetic means derived from the 12 monthly values. In the table, so far as is known, the local mean time of the observatory has been employed. This is a point requiring attention, because most observatories employ Greenwich time, or time based on Greenwich or some other national observatory, and any departure from local time enters into the values of the constants. The data for Victoria Land refer to the “Discovery’s” 1902–1903 winter quarters, where the declination, taken westerly, was about 207°.5.

As an example of the significance of the phase angles in Table XVII., take the ordinary day data for Kew. The times of occurrence of the maxima are given by t + 234° = 450° for the 24-hour term, 2t + 39°.7 = 90° or = 450° for the 12-hour term, and so on, taking an hour in t as equivalent to 15°.

Thus the times of the maxima are:—

24-hour term, 2 h. 24 m. p.m.; 12-hour term, 1 h. 41 m. a.m. and p.m.

8-hour term, 4 h. 41 m. a.m., 0 h. 41 m. p.m., and 8 h. 41 m. p.m.

6-hour term, 0 h. 33 m. a.m. and p.m., and 6 h. 33 m. a.m. and p.m.

The minima, or extreme easterly positions in the waves, lie midway between successive maxima. All four terms, it will be seen, have maxima at some hour between 0h. 30m. and 2h. 30m. p.m. They thus reinforce one another strongly from 1 to 2 p.m., accounting for the prominence of the maximum in the early afternoon.

(From Phil. Trans.)

Fig. 8.

The utility of a Fourier analysis depends largely on whether the several terms have a definite physical significance. If the 24-hour and 12-hour terms, for instance, represent the action of forces whose distribution over the earth or whose seasonal variation is essentially different, then the analysis helps to distinguish these forces, and may assist in their being tracked to their ultimate source. Suppose, for example, one had reason to think the magnetic diurnal variation due to some meteorological phenomenon, e.g. heating of the earth’s atmosphere, then a comparison of Fourier coefficients, if such existed, for the two sets of phenomena would be a powerful method of investigation.

Table XVII.—Amplitudes and Phase Angles for Diurnal Inequality of Declination.

Place.	Epoch.	c1.	c2.	c3.	c4.	α1.	α2.	α3.	α4.
		′	′	′	′	°	°	°	°
Fort Rae (all)	1882–1883	18.49	8.22	1.99	2.07	156.5	41.9	308	104
Fort Rae (quiet)	”	9.09	4.51	1.32	0.73	166.5	37.5	225	350
Ekatarinburg	1841–1862	2.57	1.81	0.73	0.22	223.3	7.4	204	351
Potsdam	1890–1899	2.81	1.90	0.83	0.31	239.9	32.6	237	49
Kew (ordinary)	1890–1900	2.91	1.79	0.79	0.27	234.0	39.7	239	57
Kew (quiet)	”	2.37	1.82	0.90	0.30	227.3	42.1	240	55
Falmouth (quiet)	1891–1902	2.18	1.82	0.91	0.29	226.2	40.5	238	56
Parc St Maur	1883–1899	2.70	1.87	0.85	0.30	238.6	32.5	235	95
Toronto	1842–1848	2.65	2.34	1.00	0.33	213.7	34.9	238	350
Washington	1840–1842	2.38	1.86	0.65	0.33	223.0	26.6	223	53
Manila	1890–1900	0.53	0.58	0.43	0.17	266.3	50.7	226	89
Trivandrum	1853–1864	0.54	0.46	0.29	0.10	289.0	49.6		114
Batavia	1883–1899	0.80	0.88	0.43	0.13	332.0	163.2	5	236
St. Helena	1842–1847	0.68	0.61	0.63	0.34	275.8	171.4	27	244
Mauritius	1876–1890	0.86	1.11	0.76	0.22	21.6	172.7	350	161
C. of G. Hope	1841–1846	1.15	1.13	0.80	0.35	287.7	156.0	351	193
Melbourne	1858–1863	2.52	2.45	1.23	0.35	27.4	176.7	9	193
Hobart	1841–1848	2.29	2.15	0.87	0.32	33.6	170.8	349	185
S. Georgia	1882–1883	2.13	1.28	0.76	0.31	30.3	185.3	7	180
Victoria Land (all)	1902–1903	20.51	4.81	1.21	1.32	158.7	306.9	292	303
Victoria Land (quieter)	”	15.34	4.05	1.24	1.18	163.8	312.9	261

§ 21. Fourier coefficients of course often vary much with the season of the year. In the case of the declination this is especially true of the phase angles at tropical stations. To enter on details for a number of stations would unduly occupy space. A fair idea of the variability in the case of declination in temperate latitudes may be derived from Table XVIII., which gives monthly values for Kew derived from ordinary days of an 11-year period 1890–1900.

Fourier analysis has been applied to the diurnal inequalities of the other magnetic elements, but more sparingly. Such results are illustrated by Table XIX., which contains data derived from quiet days at Kew from 1890 to 1900. Winter includes November to February, Summer May to August, and Equinox the remaining four months. In this case the data are derived from mean diurnal inequalities for the season specified. In the case of the c or amplitude coefficients the unit is 1′ for I (inclination), and 1γ for H and V (horizontal and vertical force). At Kew the seasonal variation in the amplitude is fairly similar for all the elements. The 24-hour and 12–hour terms tend to be largest near midsummer, and least near midwinter; but the 8-hour and 6-hour terms have two well-marked maxima near the equinoxes, and a clearly marked minimum near midsummer, in addition to one near midwinter. On the other hand, the phase angle phenomena vary much for the different elements. The 24-hour term, for instance, has its maximum earlier in winter than in summer in the case of the declination and vertical force, but the exact reverse holds for the inclination and the horizontal force.

Table XVIII.—Kew Declination: Amplitudes and Phase Angles
(local mean time).

Month.	c1.	c2.	c3.	c4.	α1.	α2.	α3.	α4.
	′	′	′	′	°	°	°	°
January	1.79	0.86	0.41	0.27	251.2	29.8	254	64
February	2.41	1.11	0.57	0.30	242.0	27.7	235	39
March	3.05	1.98	1.11	0.45	233.2	36.1	223	49
April	3.35	2.48	1.17	0.39	224.8	39.2	228	61
May	3.57	2.38	0.87	0.17	221.3	50.8	245	89
June	3.83	2.39	0.74	0.05	212.6	46.7	239	72
July	3.72	2.30	0.77	0.11	214.6	48.1	233	8
August	3.64	2.43	1.05	0.18	228.2	57.2	244	51
September	3.35	2.02	1.04	0.35	236.9	55.3	245	70
October	2.69	1.69	0.92	0.48	240.1	35.6	235	65
November	1.94	1.06	0.51	0.32	248.3	28.3	247	61
December	1.61	0.81	0.35	0.20	255.1	22.0	243	56

§ 22. If secular change proceeded uniformly throughout the year, the value E_n of any element at the middle of the nth month of the year would be connected with E, the mean value for the whole year, by the formula E_n = E + (2n − 13)s/24, where s is the secular change per annum. For the present Annual Inequality.purpose, difference in the lengths of the months may be neglected. If one applies to E_n − E the correction −(2n − 13)s/24 one eliminates a regularly progressive secular change; what remains is known as the annual inequality. If only a short period of years is dealt with, irregularities in the secular change from year to year, or errors of observation, may obviously simulate the effect of a real annual inequality. Even when a long series of years is included, there is always a possibility of a spurious inequality arising from annual variation in the instruments, or from annual change in the conditions of observation. J. Liznar,^[17] from a study of data from a number of stations, arrived at certain mean results for the annual inequalities in declination and inclination in the northern and southern hemispheres, and J. Hann^[18] has more recently dealt with Liznar’s and newer results. Table XX. gives a variety of data, including the mean results given by Liznar and Hann. In the case of declination + denotes westerly position; in the case of inclination it denotes a larger dip (whether the inclination be north or south). According to Liznar declination in summer is to the west of the normal position in both hemispheres. The phenomena, however, at Parc St Maur are, it will be seen, the exact opposite of what Liznar regards as normal; and whilst the Potsdam results resemble his mean in type, the range of the inequality there, as at Parc St Maur, is relatively small. Of the three sets of data given for Kew the first two are derived in a similar way to those for other stations; the first set are based on quiet days only, the second on all but highly disturbed days. Both these sets of results are fairly similar in type to the Parc St Maur results, but give larger ranges; they are thus even more opposed to Liznar’s normal type. The last set of data for Kew is of a special kind. During the 11 years 1890 to 1900 the Kew declination magnetograph showed to within 1′ the exact secular change as derived from the absolute observations; also, if any annual variation existed in the position of the base lines of the curves it was exceedingly small. Thus the accumulation of the daily non-cyclic changes shown by the curves should closely represent the combined effects of secular change and annual inequality. Eliminating the secular change, we arrive at an annual inequality, based on all days of the year including the highly disturbed. It is this annual inequality which appears under the heading s. It is certainly very unlike the annual inequality derived in the usual way. Whether the difference is to be wholly assigned to the fact that highly disturbed days contribute in the one case, but not in the other, is a question for future research.

Table XIX.—Kew Diurnal Inequality: Amplitudes and Phase
Angles (local mean time).

		c1.	c2.	c3.	c4.	α1.	α2.	α3.	α4.
						°	°	°	°
I	Winter	0.240	0.222	0.104	0.076	250.0	91.8	344	194
	Equinox	0.601	0.290	0.213	0.127	290.3	135.5	4	207
	Summer	0.801	0.322	0.172	0.070	312.5	155.5	39	238
H	Winter	3.62	3.86	1.81	1.13	82.9	277.3	154	6
	Equinox	10.97	5.87	3.32	1.84	109.6	303.5	167	16
	Summer	14.85	6.23	2.35	0.95	130.3	316.5	199	41
V	Winter	2.46	1.67	0.86	0.42	153.9	300.8	108	280
	Equinox	6.15	4.70	2.51	0.94	117.2	272.3	99	289
	Summer	8.63	6.45	2.24	0.55	122.0	272.4	100	285

In the case of the inclination, Liznar found that in both hemispheres the dip (north in the northern, south in the southern hemisphere) was larger than the normal when the sun was in perihelion, corresponding to an enhanced value of the horizontal force in summer in the northern hemisphere.

In the case of annual inequalities, at least that of the declination, it is a somewhat suggestive fact that the range seems to become less as we pass from older to more recent results, or from shorter to longer periods of years. Thus for Paris from 1821 to 1830 Arago deduced a range of 2′ 9″. Quiet days at Kew from 1890 to 1894 gave a range of 1′·2, while at Potsdam Lüdeling got a range 30% larger than that in Table XX. when considering the shorter period 1891–1899. Up to the present, few individual results, if any, can claim a very high degree of certainty. With improved instruments and methods it may be different in the future.

Table XX.—Annual Inequality.

Declination.

Inclination.

Liznar, N. Hemi-  sphere.

Potsdam,  1891–1906.

Parc St Maur,  1888–1897.

Kew (1890–1900).

Batavia, 1883–1893.

Mauritius.

Liznar & Hann’s mean.

Potsdam.

Parc St. Maur.

Kew.

q.

o.

s.

′

′

′

′

′

′

′

′

′

′

′

′

January

−0·25

+0·04

+0·01

+0·08

+0·03

+0·32

+0·23

+0·06

+0·49

+0·32

+0·44

−0·03

February

−0·54

−0·11

0·00

+0·48

+0·25

−0·20

+0·19

+0·29

+0·39

+0·56

+0·29

−0·07

March

−0·27

+0·04

+0·17

+0·03

+0·05

−1·02

−0·12

+0·27

+0·20

+0·38

+0·13

+0·53

April

−0·03

+0·10

+0·12

−0·31

−0·14

−0·90

−0·11

+0·30

−0·08

−0·02

−0·13

+0·18

May

+0·19

+0·07

−0·11

−0·39

−0·28

+0·29

−0·30

+0·08

−0·43

−0·29

−0·37

−0·15

June

+0·46

+0·13

−0·14

−0·47

−0·39

+0·78

−0·13

−0·19

−0·70

−0·77

−0·59

−0·35

July

+0·48

+0·14

−0·17

−0·30

−0·13

+0·44

−0·08

−0·44

−0·72

−0·67

−0·27

−0·13

August

+0·47

+0·11

+0·01

+0·08

+0·05

+0·52

−0·18

−0·38

−0·47

−0·23

−0·05

−0·19

September

+0·31

+0·01

0·00

+0·29

+0·24

−0·02

+0·06

−0·06

−0·06

+0·16

+0·01

+0·20

October

−0·07

−0·11

+0·09

+0·06

+0·01

−0·26

+0·03

−0·04

+0·31

+0·27

+0·19

0·00

November

−0·30

−0·28

−0·05

+0·17

+0·11

−0·02

+0·08

−0·01

+0·51

+0·30

+0·43

+0·18

December

−0·36

−0·14

+0·05

+0·26

+0·23

+0·05

+0·35

+0·06

+0·55

+0·19

+0·24

−0·29

Range

1·02

0·42

0·34

0·95

0·64

1·80

0·65

0·74

1·27

1·33

1·03

0·88

§ 23. The inequalities in Table XX. may be analysed—as has in fact been done by Hann—in a series of Fourier terms, whose periods are the year and its submultiples. Fourier series can also be formed representing the annual variation in the amplitudes of the regular diurnal Annual Variation Fourier Coefficients.inequality, and its component 24-hour, 12-hour, &c. waves, or of the amplitude of the absolute daily range (§ 24). To secure the highest theoretical accuracy, it would be necessary in calculating the Fourier coefficients to allow for the fact that the “months” from which the observational data are derived are not of uniform length. The mid-times, however, of most months of the year are but slightly displaced from the position they would occupy if the 12 months were exactly equal, and these displacements are usually neglected. The loss of accuracy cannot be but trifling, and the simplification is considerable.

The Fourier series may be represented by

P₁ sin (t + θ₁) + P₂ sin (2t + θ₂) + . . .,

where t is time counted from the beginning of the year, one month being taken as the equivalent of 30°, P₁, P₂ represent the amplitudes, and θ₁, θ₂ the phase angles of the first two terms, whose periods are respectively 12 and 6 months. Table XXI. gives the values of these coefficients in the case of the range of the regular diurnal inequality for certain specified elements and periods at Kew^[19] and Falmouth.^23a In the case of P₁ and P₂ the unit is 1′ for D and I, and 1γ for H and V. M denotes the mean value of the range for the 12 months. The letters q and o represent quiet and ordinary day results. S max. means the years 1892–1895, with a mean sun spot frequency of 75·0. S min. for Kew means the years 1890, 1899 and 1900 with a mean sun spot frequency of 9·6; for Falmouth it means the years 1899–1902 with a mean sun spot frequency of 7·25.

Increase in θ₁ or θ₂ means an earlier occurrence of the maximum or maxima, 1° answering roughly to one day in the case of the 12-month term, and to half a day in the case of the 6-month term. P₁/M and P₂/M both increase decidedly as we pass from years of many to years of few sun spots; i.e. relatively considered the range of the regular diurnal inequality is more variable throughout the year when sun spots are few than when they are many.

The tendency to an earlier occurrence of the maximum as we pass from quiet days to ordinary days, or from years of sun spot minimum to years of sun spot maximum, which appears in the table, appears also in the case of the horizontal force—at least in the case of the annual term—both at Kew and Falmouth. The phenomena at the two stations show a remarkably close parallelism. At both, and this is true also of the absolute ranges, the maximum of the annual term falls in all cases near midsummer, the minimum near midwinter. The maxima of the 6-month terms fall near the equinoxes.

Table XXI.—Annual Variation of Diurnal Inequality Range.
Fourier Coefficients.

		P1.	P2.	θ1.	θ2.	P1/M.	P2/M.
Kew	Do	3·36	0·94	279°	280°	0·40	0·11
1890–1900	Dq	3·81	1·22	275°	273°	0·47	0·15
	Iq	0·67	0·16	264°	269°	0·42	0·10
	Hq	13·6	3·0	269°	261°	0·48	0·11
	Vq	11·7	2·2	282°	242°	0·63	0·12
S max.	Kew	4·50	1·26	277°	282°	0·47	0·13
Dq	Falmouth	4·10	1·40	277°	286°	0·43	0·15
S min.	Kew	3·35	1·10	274°	269°	0·49	0·16
Dq	Falmouth	3·19	1·14	275°	277°	0·49	0·17

§ 24. Allusion has already been made in § 14 to one point which requires fuller discussion. If we take a European station such as Kew, the general character of, say, the declination does not vary very much with the season, but still it does vary. The principal minimum of the day, for instance, Absolute Range. occurs from one to two hours earlier in summer than in winter. Let us suppose for a moment that all the days of a month are exactly alike, the difference in type between successive months coming in per saltum. Suppose further that having formed twelve diurnal inequalities from the days of the individual months of the year, we deduce a mean diurnal inequality for the whole year by combining these twelve inequalities and taking the mean. The hours of maximum and minimum being different for the twelve constituents, it is obvious that the resulting maximum will normally be less than the arithmetic mean of the twelve maxima, and the resulting minimum (arithmetically) less than the arithmetic mean of the twelve minima. The range—or algebraic excess of the maximum over the minimum—in the mean diurnal inequality for the year is thus normally less than the arithmetic mean of the twelve ranges from diurnal inequalities for the individual months. Further, as we shall see later, there are differences in type not merely between the different months of the year, but even between the same months in different years. Thus the range of the mean diurnal inequality for, say, January based on the combined observations of, say, eleven Januarys may be and generally will be slightly less than the arithmetic mean of the ranges obtained from the Januarys separately. At Kew, for instance, taking the ordinary days of the 11 years 1890–1900, the arithmetic mean of the diurnal inequality ranges of declination from the 132 months treated independently was 8′·52, the mean range from the 12 months of the year (the eleven Januarys being combined into one, and so on) was 8′·44, but the mean range from the whole 4,000 odd days superposed was only 8′·03. Another consideration is this: a diurnal inequality is usually based on hourly readings, and the range deduced is thus an under-estimate unless the absolute maximum and minimum both happen to come exactly at an hour. These considerations would alone suffice to show that the absolute range in individual days, i.e. the difference between the algebraically largest and least values of the element found any time during the 24 hours, must on the average exceed the range in the mean diurnal inequality for the year, however this latter is formed. Other causes, moreover, are at work tending in the same direction. Even in central Europe, the magnetic curves for individual days of an ordinary month often differ widely amongst themselves, and show maxima and minima at different times of the day. In high latitudes, the variation from day to day is sometimes so great that mere eye inspection of magnetograph curves may leave one with but little idea as to the probable shape of the resultant diurnal curve for the month. Table XXII. gives the arithmetic mean of the absolute daily ranges from a few stations. The values which it assigns to the year are the arithmetic means of the 12 monthly values. The Mauritius data are for different periods, viz. declination 1875, 1880 and 1883 to 1890, horizontal force 1883 to 1890, vertical force 1884 to 1890. The other data are all for the period 1890 to 1900.

Table XXII.—Mean Absolute Daily Ranges (Units 1′ for Declination, 1γ for H and V).

	Jan.	Feb.	Mar.	April.	May.	June.	July.	Aug.	Sept.	Oct.	Nov.	Dec.	Year.
Declination.
Pavlovsk	13.42	17.20	18.22	17.25	17.76	15.91	16.89	16.57	16.75	15.70	13.87	12.37	15.99
Ekatarinburg	7.33	9.54	11.90	12.89	13.63	13.03	12.78	12.21	11.23	9.44	7.86	6.85	10.72
Kew. All days	11.16	13.69	15.93	15.00	14.90	13.65	14.13	14.22	14.57	14.07	11.71	9.80	13.57
Kew. Ordinary days	10.14	11.87	14.19	14.24	13.85	13.26	13.47	13.67	13.71	13.10	10.40	9.00	12.58
Kew. Quiet days	6.12	7.57	10.59	11.84	12.09	11.95	11.60	11.93	10.86	9.16	6.54	5.08	9.61
Zi-ka-wei	3.88	3.25	6.22	7.04	7.15	7.40	7.77	8.06	6.73	4.68	2.91	2.52	5.63
Mauritius	6.93	7.79	7.11	5.75	4.87	4.03	4.36	6.00	6.28	6.71	6.99	6.78	6.13
Horizontal force.
Pavlovsk	52.4	74.5	79.1	80.1	86.2	79.0	86.7	77.6	76.7	67.3	55.7	45.9	71.8
Ekatarinburg	33.2	43.1	48.4	51.7	56.2	54.1	56.7	51.7	49.3	44.1	34.1	29.3	46.0
Mauritius	37.9	35.0	36.2	37.6	35.0	34.1	33.8	34.5	36.6	37.4	37.8	35.3	35.9
Vertical force.
Pavlovsk	27.0	50.4	54.7	43.2	45.3	34.8	42.1	35.5	42.5	37.5	33.5	25.5	39.3
Ekatarinburg	17.4	26.6	29.2	30.1	29.6	27.6	29.6	26.1	25.2	22.1	19.6	16.4	24.9
Mauritius	17.1	19.5	20.1	17.3	16.5	15.5	17.1	22.0	22.7	19.4	16.7	15.2	18.2

A comparison of the absolute ranges in Table XXII. with the inequality ranges for the same stations derivable from Tables VIII. to X. is most instructive. At Mauritius the ratio of the absolute to the inequality range is for D 1.38, for H 1.76, and for V 1.19. At Pavlovsk the corresponding ratios are much larger, viz. 2.16 for D, 2.43 for H, and 2.05 for V. The declination data for Kew in Table XXII. illustrate other points. The first set of data are derived from all days of the year. The second omit the highly disturbed days. The third answer to the 5 days a month selected as typically quiet. The yearly mean absolute range from ordinary days at Kew in Table XXII. is 1.49 times the mean inequality range in Table VIII.; comparing individual months the ratio of the absolute to the inequality range varies from 2.06 in January to 1.21 in June. Even confining ourselves to the quiet days at Kew, which are free from any but the most trifling disturbances, we find that the mean absolute range for the year is 1.20 times the arithmetic mean of the inequality ranges for the individual months of the year, and 1.22 times the range from the mean diurnal inequality for the year. In this case the ratio of the absolute to the inequality range varies from 1.55 in December to only 1.09 in May.

§ 25. The variability of the absolute daily range of declination is illustrated by Table XXIII., which contains data for Kew^[20] derived from all days of the 11-year period 1890–1900. It gives the total number of times during the 11 years when the absolute range lay within the limits specified at the heads of the first nine columns of figures. The two remaining columns give the arithmetic means of the five largest and the five least absolute ranges encountered each month. The mean of the twelve monthly diurnal inequality ranges from ordinary days was only 8′.44, but the absolute range during the 11 years exceeded 20′ on 492 days, 15′ on 1196 days, and 10′ on 2784 days, i.e. on 69 days out of every 100.

Table XXIII.—Absolute Daily Range of Declination at Kew.

Number of occasions during 11 years when absolute range was:—										Means from the 5 largest and 5 least ranges of the month on the average of 11 years.
	0′ to 5′.	5′ to 10′.	10′ to 15′.	15′ to 20′.	20′ to 25′.	25′ to 30′.	30′ to 35′.	35′ to 40′.	over 40′.	5 largest.	5 least.
										′	′
January	51	145	69	37	24	7	4	3	1	22.90	5.07
February	26	99	84	51	26	10	4	2	8	27.21	6.55
March	1	72	138	61	32	21	8	1	7	29.87	8.93
April	0	43	167	73	27	10	6	3	1	23.69	10.31
May	0	57	157	85	20	12	3	0	7	25.36	9.50
June	0	56	185	67	15	1	3	1	2	19.92	9.89
July	0	59	185	70	14	5	2	2	4	22.49	9.96
August	0	37	202	75	22	1	2	0	2	21.27	10.05
September	1	68	153	71	19	5	4	5	4	24.55	9.52
October	3	103	111	67	34	10	11	2	0	23.92	8.01
November	42	140	81	28	14	9	8	5	3	23.58	5.64
December	64	166	56	29	14	7	1	1	3	20.43	4.36
Totals	188	1045	1588	714	261	98	56	25	42

§ 26. Magnetic phenomena, both regular and irregular, at any station vary from year to year. The extent of this variation is illustrated in Tables XXIV. and XXV., both relating to the period 1890 to 1900.^[21] Table XXIV. gives the amplitudes of Relations to Sun-spot Frequency. the regular diurnal inequality in the elements stated at the head of the columns. The ordinary day declination data (D₀) for Kew represent arithmetic means from the twelve months of the year; the other data all answer to the mean diurnal inequality for the whole year. Table XXV. gives the arithmetic means for each year of the absolute daily range, of the monthly range (or difference between the highest and lowest values in the month), and of the yearly range (or difference between the highest and lowest values of the year). The numerals attached to the years in these tables indicate their order as regards sun-spot frequency according to Wolf and Wolfer (see Aurora Polaris), 1893 being the year of largest frequency, and 1890 that of least. The difference in sun-spot frequency between 1897 and 1898 was microscopic; the differences between 1890, 1900 and 1899 were small, and those between 1893, 1894 and 1892 were not very large.

The years 1892–1895 represent high sun-spot frequency, while 1890, 1899 and 1900 represent low frequency. Table XXIV. shows that 1892 to 1895 were in all cases distinguished by the large size of the inequality ranges, and 1890, 1899 and 1900 by the small size. The range in 1893 is usually the largest, and though the H and V ranges at Ekaterinburg are larger in 1892 than in 1893, the excess is trifling. The phenomena apparent in Table XXIV. are fairly representative; other stations and other periods associate large inequality ranges with high sun-spot frequency. The diurnal inequality range it should be noticed is comparatively little influenced by irregular disturbances. Coming to Table XXV., we have ranges of a different character. The absolute range at Kew on quiet days is almost as little influenced by irregularities as is the range of the diurnal inequality, and in its case the phenomena are very similar to those observed in Table XXIV. As we pass from left to right in Table XXV., the influence of disturbance increases. Simultaneously with this, the parallelism with sun-spot frequency is less close. The entries relating to 1892 and 1894 become more and more prominent compared to those for 1893. The yearly range may depend on but a single magnetic storm, the largest disturbance of the year possibly far outstripping any other. But taking even the monthly ranges the values for 1893 are, speaking roughly, only half those for 1892 and 1894, and very similar to those of 1898, though the sun-spot frequency in the latter year was less than a third of that in 1893. Ekatarinburg data exactly analogous to those for Pavlovsk show a similar prominence in 1892 and 1894 as compared to 1893. The retirement of 1893 from first place, seen in the absolute ranges at Kew, Pavlovsk and Ekatarinburg, is not confined to the northern hemisphere. It is visible, for instance, in the amplitudes of the Batavia disturbance results. Thus though the variation from year to year in the amplitude of the absolute ranges is relatively not less but greater than that of the inequality ranges, and though the general tendency is for all ranges to be larger in years of many than in years of few sun-spots, still the parallelism between the changes in sun-spot frequency and in magnetic range is not so close for the absolute ranges and for disturbances as for the inequality ranges.

Table XXIV.—Ranges of Diurnal Inequalities.

	Pavlovsk.			Ekatarinburg.				Kew.
	D.	I.	H.	D.	I.	H.	V.	Dq.	Iq.	Hq.	Do.
	′	′	γ	′	′	γ	γ	′	′	γ	′
189011	6.32	1.33	22	5.83	1.05	18	9	6.90		20	7.32
18916	7.31	1.79	30	6.85	1.38	25	14	8.04	1.52	28	8.48
18923	8.75	2.21	37	7.74	1.72	32	19	9.50	1.66	31	9.85
18931	9.64	2.24	38	8.83	1.80	31	17	10.06	1.96	35	10.74
18942	8.58	2.17	38	7.80	1.73	30	17	9.32	1.94	34	9.80
18954	8.22	2.08	33	7.29	1.64	28	15	8.59	1.66	30	9.54
18965	7.39	1.77	29	6.50	1.38	25	15	7.77	1.31	25	8.50
18976	6.79	1.59	26	6.01	1.16	21	12	6.71	1.14	22	7.76
18987	6.25	1.56	26	5.76	1.19	21	11	6.85	1.07	21	7.59
18999	6.02	1.44	24	5.33	1.12	20	11	6.69	1.01	21	7.30
190010	6.20	1.28	22	5.88	0.93	17	8	6.52	1.06	21	6.83

Table XXV.—Absolute Ranges.

	Kew Declination.  Daily.			Pavlovsk.
				Daily.			Monthly.			Yearly.
	q.	o.	a.	D.	H.	V.	D.	H.	V.	D.	H.	V.
	′	′	′	′	γ	γ	′	γ	γ	′	γ	γ
189011	8.3	10.5	10.7	12.1	49	21	28.2	118	80	42.1	169	179
18916	10.0	12.8	13.7	16.0	70	39	46.3	218	233	92.3	550	614
18923	12.3	15.4	17.7	21.0	111	73	93.6	698	575	194.0	2416	1385
18931	11.8	15.2	15.6	17.8	79	41	48.3	241	210	87.1	514	457
18942	11.3	14.7	16.5	20.4	97	62	84.1	493	493	145.6	1227	878
18954	10.6	14.8	15.6	18.1	80	46	47.4	220	223	73.9	395	534
18965	9.5	12.9	14.5	17.5	74	43	52.4	232	236	88.7	574	608
18978	8.2	11.5	12.1	14.6	61	30	43.8	201	170	101.1	449	480
18987	8.2	11.2	12.3	14.7	67	35	46.6	276	242	118.9	1136	888
18999	7.9	10.5	11.3	13.1	58	27	38.3	178	150	63.8	382	527
190010	7.4	8.9	9.2	10.5	44	16	32.8	134	89	94.2	457	365
Means	9.6	12.6	13.6	16.0	72	39	51.1	274	246	100.2	752	629

§ 27. The relationship between magnetic ranges and sun-spot frequency has been investigated in several ways. W. Ellis^[22] has employed a graphical method which has advantages, especially for tracing the general features of the resemblance, and is besides independent of any theoretical hypothesis. Taking time for the axis of abscissae, Ellis drew two curves, one having for its ordinates the sun-spot frequency, the other the inequality range of declination or of horizontal force at Greenwich. The value assigned in the magnetic curve to the ordinate for any particular month represents a mean from 12 months of which it forms a central month, the object being to eliminate the regular annual variation in the diurnal inequality. The sun-spot data derived from Wolf and Wolfer were similarly treated. Ellis originally dealt with the period 1841 to 1877, but subsequently with the period 1878 to 1896, and his second paper gives curves representing the phenomena over the whole 56 years. This period covered five complete sun-spot periods, and the approximate synchronism of the maxima and minima, and the general parallelism of the magnetic and sun-spot changes is patent to the eye. Ellis^[23] has also applied an analogous method to investigate the relationship between sun-spot frequency and the number of days of magnetic disturbance at Greenwich. A decline in the number of the larger magnetic storms near sun-spot minimum is recognizable, but the application of the method is less successful than in the case of the inequality range. Another method, initiated by Professor Wolf of Zurich, lends itself more readily to the investigation of numerical relationships. He started by supposing an exact proportionality between corresponding changes in sun-spot frequency and magnetic range. This is expressed mathematically by the formula

R = a + bS ≡ a {1 + (b/a) S },

where R denotes the magnetic range, S the corresponding sun-spot frequency, while a and b are constants. The constant a represents the range for zero sun-spot frequency, while b/a is the proportional increase in the range accompanying unit rise in sun-spot frequency. Assuming the formula to be true, one obtains from the observed values of R and S numerical values for a and b, and can thus investigate whether or not the sun-spot influence is the same for the different magnetic elements and for different places. Of course, the usefulness of Wolf’s formula depends largely on the accuracy with which it represents the facts. That it must be at least a rough approximation to the truth in the case of the diurnal inequality at Greenwich might be inferred from Ellis’s curves. Several possibilities should be noticed. The formula may apply with high accuracy, a and b having assigned values, for one or two sun-spot cycles, and yet not be applicable to more remote periods. There are only three or four stations which have continuous magnetic records extending even 50 years back, and, owing to temperature correction uncertainties, there is perhaps no single one of these whose earlier records of horizontal and vertical force are above criticism. Declination is less exposed to uncertainty, and there are results of eye observations of declination before the era of photographic curves. A change, however, of 1′ in declination has a significance which alters with the intensity of the horizontal force. During the period 1850–1900 horizontal force in England increased about 5%, so that the force requisite to produce a declination change of 19′ in 1900 would in 1850 have produced a deflection of 20′. It must also be remembered that secular changes of declination must alter the angle between the needle and any disturbing force acting in a fixed direction. Thus secular alteration in a and b is rather to be anticipated, especially in the case of the declination. Wolf’s formula has been applied by Rajna^[24] to the yearly mean diurnal declination ranges at Milan based on readings taken twice daily from 1836 to 1894, treating the whole period together, and then the period 1871 to 1894 separately. During two sub-periods, 1837–1850 and 1854–1867, Rajna’s calculated values for the range differ very persistently in one direction from those observed; Wolf’s formula was applied by C. Chree^[21] to these two periods separately. He also applied it to Greenwich inequality ranges for the years 1841 to 1896 as published by Ellis, treating the whole period and the last 32 years of it separately, and finally to all (a) and quiet (q) day Greenwich ranges from 1889 to 1896. The results of these applications of Wolf’s formula appear in Table XXVI.

The Milan results are suggestive rather of heterogeneity in the material than of any decided secular change in a or b. The Greenwich data are suggestive of a gradual fall in a, and rise in b, at least in the case of the declination.

Table XXVII. gives values of a, b and b/a in Wolf’s formula calculated by Chree^[21] for a number of stations. There are two sets of data, the first set relating to the range from the mean diurnal inequality for the year, the second to the arithmetic mean of the ranges in the mean diurnal inequalities for the twelve months. It is specified whether the results were derived from all or from quiet days.

Table XXVI.—Values of a and b in Wolf’s Formula.

Milan.			Greenwich.
Epoch.	Declination  (unit 1′).		Epoch.	Declination  (unit 1′).		Horizontal Force  (unit 1γ).
	a.	b.		a.	b.	a.	b.
1836–94	5.31	.047	1841–96	7.29	.0377	26.4	.190
1871–94	5.39	.047	1865–96	7.07	.0396	23.6	.215
1837–50	6.43	.041	1889–96(a)	6.71	.0418	23.7	.218
1854–67	4.62	.047	1889–96(q)	6.36	.0415	25.0	.213

As explained above, a would represent the range in a year of no sun-spots, while 100 b would represent the excess over this shown by the range in a year when Wolf’s sun-spot frequency is 100. Thus b/a seems the most natural measure of sun-spot influence. Accepting it, we see that sun-spot influence appears larger at most places for inclination and horizontal force than for declination. In the case of vertical force there is at Pavlovsk, and probably in a less measure at other northern stations, a large difference between all and quiet days, which is not shown in the other elements. The difference between the values of b/a at different stations is also exceptionally large for vertical force. Whether this last result is wholly free from observational uncertainties is, however, open to some doubt, as the agreement between Wolf’s formula and observation is in general somewhat inferior for vertical force. In the case of the declination, the mean numerical difference between the observed values and those derived from Wolf’s formula, employing the values of a and b given in Table XXVII., represented on the average about 4% of the mean value of the element for the period considered, the probable error representing about 6% of the difference between the highest and lowest values observed. The agreement was nearly, if not quite, as good as this for inclination and horizontal force, but for vertical force the corresponding percentages were nearly twice as large.

Table XXVII.—Values of a and b in Wolf’s Formula.

			Declination (unit 1′).			Inclination (unit 1′).			Horizontal Force (unit 1γ).			Vertical Force (unit 1γ).
Diurnal Inequality for the Year.			a.	b.	100 b/a.	a.	b.	100 b/a.	a.	b.	100 b/a.	a.	b.	100 b/a.
Pavlovsk, 1890–1900		all	5.74	.0400	.70	1.24	.0126	1.01	20.7	.211	1.02	8.1	.265	3.26
Pavlovsk, 1890–1900		quiet	6.17	.0424	.69	· ·	· ·	· ·	20.6	.195	0.95	5.9	.027	0.46
Ekatarinburg, 1890–1900		all	5.29	.0342	.65	0.93	.0105	1.13	16.8	.182	1.09	8.6	.117	1.37
Irkutsk , 1890–1900		all	4.82	.0358	.74	0.97	.0087	0.90	18.2	.190	1.04	6.5	.071	1.09
Kew, 1890–1900		quiet	6.10	.0433	.71	0.87	.0125	1.45	18.1	.194	1.07	14.3	.081	0.56
Falmouth, 1891–1902		quiet	5.90	.0451	.76	· ·	· ·	· ·	20.1	.233	1.16	· ·	· ·	· ·
Kolaba, 1894–1901		quiet	2.37	.0066	.28	· ·	· ·	· ·	31.6	.281	0.89	19.4	.072	0.37
Batavia, 1887–1898		all	2.47	.0179	.72	3.60	.0218	0.61	38.7	.274	0.71	30.1	.156	0.52
Mauritius	1875–1880 1883–1890	all	4.06	.0164	.40	· ·	· ·	· ·	15.0	.096	0.64	11.9	.069	0.58
Mean from individual months:—
Pavlovsk, 1890–1900		all	6.81	.0446	.66	1.44	.0151	1.05	22.8	.243	1.07	9.7	.287	2.97
Pavlovsk, 1890–1900		quiet	6.52	.0442	.68	· ·	· ·	· ·	22.2	.208	0.94	7.0	.044	0.63
Ekatarinburg, 1890–1900		all	6.18	.0355	.58	1.12	.0120	1.06	19.2	.195	1.01	9.2	.156	1.70
Greenwich, 1865–1896		all	7.07	.0396	.56	· ·	· ·	· ·	23.6	.215	0.91	· ·	· ·	· ·
Kew, 1890–1900		all	6.65	.0428	.64	· ·	· ·	· ·	· ·	· ·	· ·	· ·	· ·	· ·
Kew, 1890–1900		quiet	6.49	.0410	.63	1.17	.0130	1.11	21.5	.191	0.89	16.0	.072	0.45
Falmouth, 1891–1902		quiet	6.16	.0450	.73	· ·	· ·	· ·	20.9	.236	1.13	· ·	· ·	· ·

Applying Wolf’s formula to the diurnal ranges for different months of the year, Chree found, as was to be anticipated, that the constant a had an annual period, with a conspicuous minimum at midwinter; but whilst b also varied, it did so to a much less extent, the consequence being that b/a showed a minimum at midsummer. The annual variation in b/a alters with the place, with the element, and with the type of day from which the magnetic data are derived. Thus, in the case of Pavlovsk declination, whilst the mean value of 100 b/a for the 12 months is, as shown in Table XXVII., 0.66 for all and 0.68 for quiet days—values practically identical—if we take the four midwinter and the four midsummer months separately, we have 100 b/a, varying from 0.81 in winter to 0.52 in summer on all days, but from 1.39 in winter to 0.52 in summer on quiet days. In the case of horizontal force at Pavlovsk the corresponding figures to these are for all days—winter 1.77, summer 0.98, but for quiet days—winter 1.83, summer 0.71.

Wolf’s formula has also been applied to the absolute daily ranges, to monthly ranges, and to various measures of disturbance. In these cases the values found for b/a are usually larger than those found for diurnal inequality ranges, but the accordance between observed values and those calculated from Wolf’s formula is less good. If instead of the range of the diurnal inequality we take the sum of the 24-hourly differences from the mean for the day—or, what comes to the same thing, the average departure throughout the 24 hours from the mean value for the day—we find that the resulting Wolf’s formula gives at least as good an agreement with observation as in the case of the inequality range itself. The formulae obtained in the case of the 24 differences, at places as wide apart as Kew and Batavia, agreed in giving a decidedly larger value for b/a than that obtained from the ranges. This indicates that the inequality curve is relatively less peaked in years of many than in years of few sun-spots.

§ 28. The applications of Ellis’s and Wolf’s methods relate directly only to the amplitude of the diurnal changes. There is, however, a change not merely in amplitude but in type. This is clearly seen when we compare the values found in years of many and of few sun-spots for the Fourier coefficients in the diurnal inequality. Such a comparison is carried out in Table XXVIII. for the declination on ordinary days at Kew. Local mean time is used. The heading S max. (sun-spot maximum) denotes mean average results from the four years 1892–1895, having a mean sun-spot frequency of 75.0, whilst S min. (sun-spot minimum) applies similarly to the years 1890, 1899 and 1900, having a mean sun-spot frequency of only 9.6. The data relate to the mean diurnal inequality for the whole year or for the season stated. It will be seen that the difference between the c, or amplitude, coefficients in the S max. and S min. years is greater for the 24-hour term than for the 12-hour term, greater for the 12-hour than for the 8-hour term, and hardly apparent in the 6-hour term. Also, relatively considered, the difference between the amplitudes in S max. and S min. years is greatest in winter and least in summer. Except in the case of the 6-hour term, where the differences are uncertain, the phase angle is larger, i.e. maxima and minima occur earlier in the day, in years of S min. than in years of S max. Taking the results for the whole year in Table XXVIII., this advance of phase in the S min. years represents in time 15.6 minutes for the 24–hour term, 9.4 minutes for the 12-hour term, and 14.7 minutes for the 8-hour term. The difference in the phase angles, as in the amplitudes, is greatest in winter. Similar phenomena are shown by the horizontal force, and at Falmouth^[20] as well as Kew.

Table XXVIII.—Fourier Coefficients in Years of many and few Sun-spots.

	Year.		Winter.		Equinox.		Summer.
	S max.	S min.	S max.	S min.	S max.	S min.	S max.	S min.
	′	′	′	′	′	′	′	′
c1	3.47	2.21	2.41	1.43	3.76	2.41	4.38	2.98
c2	2.04	1.51	1.15	0.78	2.33	1.71	2.73	2.06
c3	0.89	0.72	0.55	0.42	1.16	0.97	0.97	0.77
c4	0.28	0.27	0.30	0.27	0.42	0.42	0.11	0.11
	°	°	°	°	°	°	°	°
α1	228.5	232.4	243.0	256.0	231.3	233.7	218.2	220.3
α2	41.7	46.6	23.5	36.9	40.6	43.9	50.6	52.5
α3	232.6	243.6	234.0	257.6	228.4	236.2	236.8	245.4
α4	58.0	57.3	52.3	60.8	62.0	58.2	57.4	45.2

§ 29. There have already been references to quiet days, for instance in the tables of diurnal inequalities. It seems to have been originally supposed that quiet days differed from other days only in the absence of irregular disturbances, and that mean Quiet Day Phenomena. annual values, or secular change data, or diurnal inequalities, derived from them might be regarded as truly normal or representative of the station. It was found, however, by P. A. Müller^[25] that mean annual values of the magnetic elements at St Petersburg and Pavlovsk from 1873 to 1885 derived from quiet days alone differed in a systematic fashion from those derived from all days, and analogous results were obtained by Ellis^[26] at Greenwich for the period 1889–1896. The average excesses for the quiet-day over the all-day means in these two cases were as follows:—

	Westerly  Declination.	Inclination.	Horizontal  Force.	Vertical  Force.
St Petersburg	+0.24	−0.23	+3.2γ	−0.8γ
Greenwich	+0.08		+3.2γ	−0.9γ

The sign of the difference in the case of D, I and H was the same in each year examined by Müller, and the same was true of H at Greenwich. In the case of V, and of D at Greenwich, the differences are

small and might be accidental. In the case of D at Greenwich 1891 differed from the other years, and of two more recent years examined by Ellis^[27] one, 1904, agreed with 1891. At Kew, on the average of the 11 years 1890 to 1900, the quiet-day mean annual value of declination exceeded the ordinary day value, but the apparent excess 0′.02 is too small to possess much significance.

Another property more recently discovered in quiet days is the non-cyclic change. The nature of this phenomenon will be readily understood from the following data from the 11-year period 1890 to 1900 at Kew^[28]. The mean daily change for all days is calculated from the observed annualNon-cyclic Change. change.

	D.	I.	H.	V.
	′	′
Mean annual change	−5.79	−2.38	+25.9γ	−22.6γ
Mean daily change, all days	−0.016	−0.007	+0.07γ	−0.06γ
Mean daily change, quiet days	+0.044	−0.245	+3.34γ	−0.84γ

Thus the changes during the representative quiet day differed from those of the average day. Before accepting such a phenomenon as natural, instrumental peculiarities must be carefully considered. The secular change is really based on the absolute instruments, the diurnal changes on the magnetographs, and the first idea likely to occur to a critical mind is that the apparent abnormal change on quiet days represents in reality change of zero in the magnetographs. If, however, the phenomenon were instrumental, it should appear equally on days other than quiet days, and we should thus have a shift of zero amounting in a year to over 1,200γ in H, and to about 90′ in I. Under such circumstances the curve would be continually drifting off the sheet. In the case of the Kew magnetographs, a careful investigation showed that if any instrumental change occurred in the declination magnetograph during the 11 years it did not exceed a few tenths of a minute. In the case of the H and V magnetographs at Kew there is a slight drift, of instrumental origin, due to weakening of the magnets, but it is exceedingly small, and in the case of H is in the opposite direction to the non-cyclic change on quiet days. It only remains to add that the hypothesis of instrumental origin was positively disproved by measurement of the curves on ordinary days.

It must not be supposed that every quiet day agrees with the average quiet day in the order of magnitude, or even in the sign, of the non-cyclic change. In fact, in not a few months the sign of the non-cyclic change on the mean of the quiet days differs from that obtained for the average quiet day of a period of years. At Kew, between 1890 and 1900, the number of months during which the mean non-cyclic change for the five quiet days selected by the astronomer royal (Sir W. H. M. Christie) was plus, zero, or minus, was as follows:—

Element.	D.	I.	H.	V.
Number +	63	13	112	47
Number 0	14	16	11	9
Number −	55	101	9	74

The + sign denotes westerly movement in the declination, and increasing dip of the north end of the needle. In the case of I and H the excess in the number of months showing the normal sign is overwhelming. The following mean non-cyclic changes on quiet days are from other sources:—

Element.	Greenwich  (1890–1895).	Falmouth  (1898–1902).	Kolaba  (1894–1901).
	′	′	′
D	+ 0.03	+ 0.05	+ 0.07
H	+ 4.3γ	+ 3.0γ	+ 3.9γ

The results are in the same direction as at Kew, + meaning in the case of D movement to the west. At Falmouth^[28], as at Kew, the non-cyclic change showed a tendency to be small in years of few sun-spots.

§ 30. In calculating diurnal inequalities from quiet days the non-cyclic effect must be eliminated, otherwise the result would depend on the hour at which the “day” is supposed to commence. If the value recorded at the second midnight of the average day exceeds that at the first midnight by N, the elimination is effected by applying to each hourly value the correction N(12 − n)/24, where n is the hour counted from the first midnight (0 hours). This assumes the change to progress uniformly throughout the 24 hours. Unless this is practically the case—a matter difficult either to prove or disprove—the correction may not secure exactly what is aimed at. This method has been employed in the previous tables. The fact that differences do exist between diurnal inequalities derived from quiet days and all ordinary days was stated explicitly in § 4, and is obvious in Tables VIII. to XI. An extreme case is represented by the data for Jan Mayen in these tables. Figs. 9 and 10 are vector diagrams for this station, for all and for quiet days during May, June and July 1883, according to data got out by Lüdeling. As shown by the arrows, fig. 10 (quiet days) is in the main described in the normal or clockwise direction, but fig. 9 (all days) is described in the opposite direction. Lüdeling found this peculiar difference between all and quiet days at all the north polar stations occupied in 1882–1883 except Kingua Fjord, where both diagrams were described clockwise.

Fig. 9.	Fig. 10.

In temperate latitudes the differences of type are much less, but still they exist. A good idea of their ordinary size and character in the case of declination may be derived from Table XXIX., containing data for Kew, Greenwich and Parc St Maur.

The data for Greenwich are due to W. Ellis^[26], those for Parc St Maur to T. Moureaux^[29]. The quantity tabulated is the algebraic excess of the all or ordinary day mean hourly value over the corresponding quiet day value in the mean diurnal inequality for the year. At Greenwich and Kew days of extreme disturbance have been excluded from the ordinary days, but apparently not at Parc St Maur. The number of highly disturbed days at the three stations is, however, small, and their influence is not great. The differences disclosed by Table XXIX. are obviously of a systematic character, which would not tend to disappear however long a period was utilized. In short, while the diurnal inequality from quiet days may be that most truly representative of undisturbed conditions, it does not represent the average state of conditions at the station. To go into full details respecting the differences between all and quiet days would occupy undue space, so the following brief summary of the differences observed in declination at Kew must suffice. While the inequality range is but little different for the two types of days, the mean of the hourly differences from the mean for the day is considerably reduced in the quiet days. The 24-hour term in the Fourier analysis is of smaller amplitude in the quiet days, and its phase angle is on the average about 6°.75 smaller than on ordinary days, implying a retardation of about 27 minutes in the time of maximum. The diurnal inequality range is more variable throughout the year in quiet days than on ordinary days, and the same is true of the absolute ranges. The tendency to a secondary minimum in the range at midsummer is considerably more decided on ordinary than on quiet days. When the variation throughout the year in the diurnal inequality range is expressed in Fourier series, whose periods are the year and its submultiples, the 6-month term is notably larger for ordinary than for quiet days. Also the date of the maximum in the 12-month term is about three days earlier for ordinary than for quiet days. The exact size of the differences between ordinary and quiet day phenomena must depend to some extent on the criteria employed in selecting quiet days and in excluding disturbed days. This raises difficulties when it comes to comparing results at different stations. For stations near together the difficulty is trifling. The astronomer royal’s quiet days have been used for instance at Parc St. Maur, Val Joyeux, Falmouth and Kew, as well as at Greenwich. But when stations are wide apart there are two obvious difficulties: first, the difference of local time; secondly, the fact that a day may be typically quiet at one station but appreciably disturbed at the other.

If the typical quiet day were simply the antithesis of a disturbed day, it would be natural to regard the non-cyclic change on quiet days as a species of recoil from some effect of disturbance. This view derives support from the fact, pointed out long ago by Sabine^[30], that the horizontal force usually, though by no means always, is lowered by magnetic disturbances. Dr van Bemmelen^[31] who has examined non-cyclic phenomena at a number of stations, seems disposed to regard this as a sufficient explanation. There are, however, difficulties in accepting this view. Thus, whilst the non-cyclic effect in horizontal force and inclination at Kew and Falmouth appeared on the whole enhanced in years of sun-spot maximum, the difference between years such as 1892 and 1894 on the one hand, and 1890 and 1900 on the other, was by no means proportional to the excess of disturbance in the former years. Again, when the average non-cyclic change of declination was calculated at Kew for 207 days, selected as those of most marked irregular disturbance between 1890 and 1900, the sign actually proved to be the same as for the average quiet day of the period.

Table XXIX.—All or Ordinary, less Quiet Day Hourly Values (+ to the West).

Hour.	Forenoon.			Afternoon.
	Kew  1890–1900.	Greenwich  1890–1894.	Parc St Maur  1883–1897.	Kew  1890–1900.	Greenwich  1890–1894.	Parc St Maur  1893–1897.
	′	′	′	′	′	′
1	−0.58	−0.59	−0.63	+0.42	+0.44	+0.40
2	−0.54	−0.47	−0.47	+0.52	+0.45	+0.50
3	−0.51	−0.31	−0.32	+0.57	+0.52	+0.59
4	−0.41	−0.23	−0.16	+0.60	+0.51	+0.55
5	−0.28	−0.10	−0.01	+0.46	+0.34	+0.38
6	−0.08	+0.12	+0.18	+0.21	+0.04	+0.07
7	+0.13	+0.30	+0.34	−0.06	−0.24	−0.25
8	+0.29	+0.48	+0.47	−0.27	−0.50	−0.54
9	+0.40	+0.56	+0.53	−0.47	−0.68	−0.74
10	+0.44	+0.58	+0.51	−0.61	−0.78	−0.79
11	+0.48	+0.50	+0.44	−0.62	−0.77	−0.79
12	+0.45	+0.44	+0.38	−0.54	−0.61	−0.67

§ 31. A satisfactory definition of magnetic disturbance is about as difficult to lay down as one of heterodoxy. The idea in its generality seems to present no difficulty, but it is a very different matter when one comes to details. Amongst the chief disturbances recorded since 1890 are those of Magnetic Disturbances. February 13–14 and August 12, 1892; July 20 and August 20, 1894; March 15–16, and September 9, 1898; October 31, 1903; February 9–10, 1907; September 11–12, 1908 and September 25, 1909. On such days as these the oscillations shown by the magnetic curves are large and rapid, aurora is nearly always visible in temperate latitudes, earth currents are prominent, and there is interruption—sometimes very serious—in the transmission of telegraph messages both in overhead and underground wires. At the other end of the scale are days on which the magnetic curves show practically no movement beyond the slow regular progression of the regular diurnal inequality. But between these two extremes there are an infinite variety of intermediate cases. The first serious attempt at a precise definition of disturbance seems due to General Sabine^[31]. His method had once an extensive vogue, and still continues to be applied at some important observatories. Sabine regarded a particular observation as disturbed when it differed from the mean of the observations at that hour for the whole month by not less than a certain limiting value. His definition takes account only of the extent of the departure from the mean, whether the curve is smooth at the time or violently oscillating makes no difference. In dealing with a particular station Sabine laid down separate limiting values for each element. These limits were the same, irrespective of the season of the year or of the sun-spot frequency. A departure, for example, of 3′.3 at Kew from the mean value of declination for the hour constituted a disturbance, whether it occurred in December in a year of sun-spot minimum, or in June in a year of sun-spot maximum, though the regular diurnal inequality range might be four times as large in the second case as in the first. The limiting values varied from station to station, the size depending apparently on several considerations not very clearly defined. Sabine subdivided the disturbances in each element into two classes: the one tending to increase the element, the other tending to diminish it. He investigated how the numbers of the two classes varied throughout the day and from month to month. He also took account of the aggregate value of the disturbances of one sign, and traced the diurnal and annual variations in these aggregate values. He thus got two sets of diurnal variations and two sets of annual variations of disturbance, the one set depending only on the number of the disturbed hours, the other set considering only the aggregate value of the disturbances. Generally the two species of disturbance variations were on the whole fairly similar. The aggregates of the + and − disturbances for a particular hour of the day were seldom equal, and thus after the removal of the disturbed values the mean value of the element for that hour was generally altered. Sabine’s complete scheme supposed that after the criterion was first applied, the hourly means would be recalculated from the undisturbed values and the criterion applied again, and that this process would be repeated until the disturbed observations all differed by not less than the accepted limiting value from the final mean based on undisturbed values alone. If the disturbance limit were so small that the disturbed readings formed a considerable fraction of the whole number, the complete execution of Sabine’s scheme would be exceedingly laborious. As a matter of fact, his disturbed readings were usually of the order of 5% of the total number, and unless in the case of exceptionally large magnetic storms it is of little consequence whether the first choice of disturbed readings is accepted as final or is reconsidered in the light of the recalculated hourly means.

Sabine applied his method to the data obtained during the decade 1840 to 1850 at Toronto, St Helena, Cape of Good Hope and Hobart, also to data for Pekin, Nertchinsk, Point Barrow, Port Kennedy and Kew. C. Chambers^[32] applied it to data from Bombay. The yearly publication of the Batavia observatory gives corresponding results for that station, and Th. Moureaux ^[29] has published similar data for Parc St Maur. Tables XXX. to XXXII. are based on a selection of these data. Tables XXX. and XXXI. show the annual variation in Sabine’s disturbances, the monthly values being expressed as percentages of the arithmetic mean value for the 12 months. The Parc St Maur and Batavia data, owing to the long periods included, are especially noteworthy. Table XXX. deals with the east (E) and west (W) disturbances of declination separately. Table XXXI., dealing with disturbances in horizontal and vertical force, combines the + and − disturbances, treated numerically. At Parc St Maur the limits required to qualify for disturbance were 3′.0 in D, 20γ in H, and 12γ in V; the corresponding limits for Batavia were 1′.3, 11γ and 11γ. The limits for D at Toronto, Bombay and Hobart were respectively 3′.6, 1′.4 and 2′.4.

At Parc St Maur the disturbance data from all three elements give distinct maxima near the equinoxes; a minimum at midwinter is clearly shown, and also one at midsummer, at least in D and H. A decline in disturbance at midwinter is visible at all the stations, but at Batavia the equinoctial values for D and V are inferior to those at midsummer.

Table XXX.—Annual Variation of Disturbances
(Sabine’s numbers).

	Parc St Maur  1883–97.		Toronto 1841–48.		Bombay 1859–65.		Batavia 1883–99.		Hobart 1843–48.
Month.	E.	W.	E.	W.	E.	W.	E.	W.	E.	W.
January	78	60	55	66	89	89	180	223	165	182
February	116	92	75	86	94	67	138	144	121	116
March	126	107	92	94	129	97	102	87	114	104
April	105	113	115	114	106	129	67	73	110	102
May	101	118	101	101	63	99	72	71	62	53
June	77	89	95	72	78	81	45	27	32	37
July	82	104	140	126	121	173	62	46	50	49
August	88	113	137	133	154	131	69	69	86	78
September	134	137	163	139	111	108	135	144	135	114
October	119	115	101	111	140	128	95	88	124	123
November	99	94	73	85	43	43	106	91	79	111
December	75	58	51	72	72	55	124	137	123	130

Table XXXII. shows in some cases a most conspicuous diurnal variation in Sabine’s disturbances. The data are percentages of the totals for the whole 24 hours. But whilst at Batavia the easterly and westerly disturbances in D vary similarly, at Parc St Maur they follow opposite laws, the easterly showing a prominent maximum near noon, the westerly a still more prominent maximum near midnight. The figures in the second last line of the table, if divided by 0.24, will give the percentage of hours which show the species of disturbance indicated. For instance, at Parc St Maur, out of 100 hours, 3 show disturbances to the west and 3.7 to the east; or in all 6.7 show disturbances of declination. The last line gives the average size of a disturbance of each type, the unit being 1′ in D and 1γ in H and V.

Table XXXI.—Annual Variation of Disturbances.

	Parc St Maur.		Toronto.		Batavia.
Month.	Numbers.		Aggregates.		Numbers.		Aggregates.
	H.	V.	H.	V.	H.	V.	H.	V.
January	81	51	58	56	96	151	89	154
February	96	133	94	74	105	123	110	125
March	126	118	94	108	116	105	117	103
April	94	111	150	149	104	76	105	73
May	108	133	90	112	101	92	105	95
June	90	85	36	50	82	69	79	66
July	99	128	61	71	90	83	95	81
August	113	92	75	108	91	91	98	91
September	119	122	171	160	113	111	114	115
October	101	94	148	129	114	89	104	86
November	104	81	98	75	99	102	100	101
December	70	51	128	100	89	108	84	110

At Batavia disturbances increasing and decreasing the element are about equally numerous, but this is exceptional. Easterly disturbances of declination predominated at Toronto, Point Barrow, Fort Kennedy, Kew, Parc St Maur, Bombay and the Falkland Islands whilst the reverse was true of St Helena, Cape of Good Hope, Pekin and Hobart. At Kew and Parc St Maur the ratios borne by the eastern to the western disturbances were 1.19 and 1.23 respectively, and so not much in excess of unity; but the preponderance of easterly disturbances at the North American^[33] stations was considerably larger than this.

Table XXXII.—Diurnal Variation of Disturbances (Sabine’s numbers).

Hour.	Parc St Maur.						Batavia.
	D.		H.		V.		D.		H.		V.
	E.	W.	+	−	+	−	E.	W.	+	−	+	−
0–3	10.1	20.3	9.0	8.3	5.7	9.2	1.1	5.8	13.1	6.6	4.0	7.4
3–6	12.3	8.2	8.4	8.0	6.4	10.4	7.6	7.3	14.2	4.8	6.3	10.0
6–9	15.7	3.8	14.1	12.5	7.2	9.0	24.9	16.8	12.1	9.9	21.2	21.7
9–noon	16.2	5.1	18.0	15.6	12.9	15.4	38.5	33.0	8.6	15.8	19.8	16.4
noon–3	19.3	6.7	15.3	16.5	18.2	18.3	18.8	24.7	16.8	21.1	23.5	22.1
3–6	14.8	9.7	12.5	15.4	22.9	21.8	6.4	5.4	13.3	16.9	12.6	12.7
6–9	5.7	21.2	11.4	13.2	18.9	11.2	2.3	3.4	9.9	13.6	7.1	4.1
9–12	5.9	25.0	11.2	10.5	7.8	4.7	0.4	3.8	12.0	11.1	5.6	5.4
Mean number per day${\Big \}}$	0.88	0.72	1.15	1.56	1.04	0.96	0.46	0.44	1.62	1.61	1.19	1.13
Mean size	· ·	· ·	· ·	· ·	· ·	· ·	1.72	1.69	18.0	19.5	16.7	15.5

§ 32. From the point of view of the surveyor there is a good deal to be said for Sabine’s definition of disturbance, but it is less satisfactory from other standpoints. One objection has been already indicated, viz. the arbitrariness of applying the same limiting value at a station irrespective of the size of the normal diurnal range at the time. Similarly it is arbitrary to apply the same limit between 10 a.m. and noon, when the regular diurnal variation is most rapid, as between 10 p.m. and midnight, when it is hardly appreciable. There seems a distinct difference of phase between the diurnal inequalities on different types of days at the same season; also the phase angles in the Fourier terms vary continuously throughout the year, and much more rapidly at some stations and at some seasons than at others. Thus there may be a variety of phenomena which one would hesitate to regard as disturbances which contribute to the annual and diurnal variations in Tables XXX. to XXXII.

Sabine, as we have seen, confined his attention to the departure of the hourly reading from the mean for that hour. Another and equally natural criterion is the apparent character of the magnetograph curve. At Potsdam curves are regarded as “1” quiet, “2” moderately disturbed, or “3” highly disturbed. Any hourly value to which the numeral 3 is attached is treated as disturbed, and the annual Potsdam publication contains tables giving the annual and diurnal variations in the number of such disturbed hours for D, H and V. According to this point of view, the extent to which the hourly value departs from the mean for that hour is immaterial to the results. It is the greater or less sinuosity and irregularity of the curve that counts. Tables XXXIII. and XXXIV. give an abstract of the mean Potsdam results from 1892 to 1901. The data are percentages: in Table XXXIII. of the mean monthly total, in Table XXXIV. of the total for the day. So far as the annual variation is concerned, the results in Table XXXIII. are fairly similar to those in Table XXX. for Parc St Maur. There are pronounced maxima near the equinoxes, especially the spring equinox. The diurnal variations, however, in Tables XXXII. and XXXIV. are dissimilar. Thus in the case of H the largest disturbance numbers at Parc St Maur occurred between 6 a.m. and 6 p.m., whereas in Table XXXIV. they occur between 4 p.m. and midnight. Considering the comparative proximity of Parc St Maur and Potsdam, one must conclude that the apparent differences between the results for these two stations are due almost entirely to the difference in the definition of disturbance.

Table XXXIII.—Annual Variation of Potsdam Disturbances.

Element.	Jan.	Feb.	Mar.	April.	May.	June.	July.	Aug.	Sept.	Oct.	Nov.	Dec.
D	129	170	149	90	86	57	62	64	59	118	94	82
H	109	133	131	102	109	82	94	91	89	101	75	84
V	106	171	170	108	121	56	64	74	93	87	78	70
Mean	115	158	150	100	105	65	73	76	94	102	82	79

Table XXXIV.—Diurnal Variation of Potsdam Disturbances.

Hours.	13.	4–6.	7–9.	10–noon.	1–3.	4–6.	7–9.	10–12.
D	14.9	11.1	8.0	5.2	5.7	13.1	22.5	19.5
H	10.5	8.4	8.0	8.5	11.3	17.6	19.2	16.5
V	13.5	9.7	5.7	4.7	8.5	17.2	21.5	19.2
Mean	13.0	9.7	7.2	6.1	8.5	16.0	21.1	18.4

Table XXXV.—Disturbed Day less ordinary Day Inequality (Unit 1′, + to West).

Hour.

1

2

3

4

5

6

7

8

9

10

11

12

a.m.

−3.4

−2.6

−2.0

−0.3

+1.6

+1.9

+2.3

+2.0

+2.1

+2.0

+1.6

+1.8

p.m.

+1.8

+2.2

+2.1

+1.7

+1.4

0.0

−1.3

−2.8

−3.5

−2.6

−3.5

−2.4

One difficulty in the Potsdam procedure is the maintenance of a uniform standard. Unless very frequent reference is made to the curves of some standard year there must be a tendency to enter under “3” in quiet years a number of hours which would be entered under “2” in a highly disturbed year. Still, such a source of uncertainty is unlikely to have much influence on the diurnal, or even on the annual, variation.

§ 33. A third method of investigating a diurnal period in disturbances is to form a diurnal inequality from disturbed days alone, and compare it with the corresponding inequalities from ordinary or from quiet days. Table XXXV. gives some declination data for Kew, the quantity tabulated being the algebraic excess of the disturbed day hourly value over that for the ordinary day in the mean diurnal inequality for the year, as based on the 11 years 1890 to 1900.

Fig. 11.

The disturbed day inequality was corrected for non-cyclic change in the usual way. Fig. 11 shows the results of Table XXXV. graphically. The irregularities are presumably due to the limited number, 209, of disturbed days employed; to get a smooth curve would require probably a considerably longer period of years. The differences between disturbed and ordinary days at Kew are of the same general character as those between ordinary and quiet days in Table XXIX.; they are, however, very much larger, the range in Table XXXV. being fully 51/2 times that in Table XXIX. If quiet days had replaced ordinary days in Table XXXV., the algebraic excess of the disturbed day would have varied from +2′.7 at 2 p.m. to −4′.1 at 11 p.m., or a range of 6′.8.

§ 34. When the mean diurnal inequality in declination for the year at Kew is analysed into Fourier waves, the chief difference, it will be remembered, between ordinary and quiet days was that the amplitude of the 24-hour term was enhanced in the ordinary days, whilst its phase angle indicated an earlier occurrence of the maximum. Similarly, the chief difference between the Fourier waves for the disturbed and ordinary day inequalities at Kew is the increase in the amplitude of the 24-hour term in the former by over 70%, and the earlier occurrence of its maximum by about 1 hour 50 minutes. It is clear from these results for Kew, and it is also a necessary inference from the differences obtained by Sabine’s method between east and west or + and − disturbances, that there is present during disturbances some influence which affects the diurnal inequality in a regular systematic way, tending to make the value of the element higher during some hours and lower during others than it is on days relatively free from disturbance. At Kew the consequence is a notable increase in the range of the regular diurnal inequality on disturbed days; but whether this is the general rule or merely a local peculiarity is a subject for further research.

§ 35. There are still other ways of attacking the problem of disturbances. W. Ellis^[23] made a complete list of disturbed days at Greenwich from 1848 onwards, arranging them in classes according to the amplitude of the disturbance shown on the curves. Of the 18,000 days which he considered, Ellis regarded 2,119, or only about 12%, as undisturbed. On 11,898 days, or 66%, the disturbance movement in declination was under 10′; on 3614, or 20%, the disturbance, though exceeding 10′, was under 30′; on 294 days it lay between 30′ and 60′; while on 75 days it exceeded 60′. Taking each class of disturbances separately, Ellis found, except in the case of his “minor” disturbances—those under 10′—a distinct double annual period, with maxima towards the equinoxes. Subsequently C. W. Maunder,^[34] making use of these same data, and of subsequent data up to 1902, put at his disposal by Ellis, came to similar conclusions. Taking all the days with disturbances of declination over 10′, and dealing with 15-day periods, he found the maxima of frequency to occur the one a little before the spring equinox, the other apparently after the autumnal equinox; the two minima were found to occur early in June and in January. When the year is divided into three seasons—winter (November to February), summer (May to August), and equinox—Maunder’s figures lead to the results assigned to Greenwich disturbed days in Table XXXVI. The frequency in winter, it will be noticed, though less than at equinox, is considerably greater than in summer. This greater frequency in winter is only slightly apparent in the disturbances over 60′, but their number is so small that this may be accidental. The next figures in Table XXXVI. relate to highly disturbed days at Kew. The larger relative frequency at Kew in winter as compared to summer probably indicates no real difference from Greenwich, but is simply a matter of definition. The chief criterion at Kew for classifying the days was not so much the mere amplitude of the largest movement, as the general character of the day’s curve and its departure from the normal form. The data in Table XXXVI. as to magnetic storms at Greenwich are based on the lists given by Maunder^[35] in the Monthly Notices, R.A.S. A storm may last for any time from a few hours to several days, and during part of its duration the disturbance may not be very large; thus it does not necessarily follow that the frequencies of magnetic storms and of disturbed days will follow the same laws. The table shows, however, that so far as Greenwich is concerned the annual variations in the two cases are closely alike. In addition to mean data for the whole 56 years, 1848 to 1903, Table XXXVI. contains separate data for the 14 years of that period which represented the highest sun-spot frequency, and the 15 years which represented lowest sun-spot frequency. It will be seen that relatively considered the seasonal frequencies of disturbance are more nearly equal in the years of many than in those of few sun-spots. Storms are more numerous as a whole in the years of many sun-spots, and this preponderance is especially true of storms of the largest size. This requires to be borne in mind in any comparisons between larger and smaller storms selected promiscuously from a long period. An unduly large proportion of the larger storms will probably come from years of large sun-spot frequency, and there is thus a risk of assigning to differences between the laws obeyed by large and small storms phenomena that are due in whole or in part to differences between the laws followed in years of many and of few sun-spots. The last data in Table XXXVI. are based on statistics for Batavia given by W. van Bemmelen,^[36] who considers separately the storms which commence suddenly and those which do not. These sudden movements are recorded over large areas, sometimes probably all over the earth, if not absolutely simultaneously, at least too nearly so for differences in the time of occurrence to be shown by ordinary magnetographs. It is ordinarily supposed that these sudden movements, and the storms to which they serve as precursors, arise from some source extraneous to the earth, and that the commencement of the movement intimates the arrival, probably in the upper atmosphere, of some form of energy transmitted through space. In the storms which commence gradually the existence of a source external to the earth is not so prominently suggested, and it has been sometimes supposed that there is a fundamental difference between the two classes of storms. Table XXXVI. shows, however, no certain difference in the annual variation at Batavia. At the same time, this possesses much less significance than it would have if Batavia were a station like Greenwich, where the annual variation in magnetic storms is conspicuous.

Besides the annual period, there seems to be also a well-marked diurnal period in magnetic disturbances. This is apparent in Tables XXXVII. and XXXVIII., which contain some statistics for Batavia due to van Bemmelen, and some for Greenwich derived from the data in Maunder’s papers referred to above. Table XXXVII. gives the relative frequency of occurrence for two hour intervals, starting with midnight, treating separately the storms of gradual (g) and sudden (s) commencement. In Table XXXVIII. the day is subdivided into three equal parts. Batavia and Greenwich agree in showing maximum frequency of beginnings about the time of minimum frequency of endings and conversely; but the hours at which the respective maxima and minima occur at the two places differ rather notably.

§ 36. There are peculiarities in the sudden movements ushering in magnetic storms which deserve fuller mention. According to van Bemmelen the impulse consists usually at some stations of a sudden slight jerk of the magnet in one direction, followed by a larger decided movement in the opposite direction, the former being often indistinctly shown. Often we have at the very commencement but a faint outline, and thereafter a continuous movement which is only sometimes distinctly indicated, resulting after some minutes in the displacement of the trace by a finite amount from the position it occupied on the paper before the disturbance began. This may mean, as van Bemmelen supposes, a small preliminary movement in the opposite direction to the clearly shown displacement; but it may only mean that the magnet is initially set in vibration, swinging on both sides of the position of equilibrium, the real displacement of the equilibrium position being all the time in the direction of the displacement apparent after a few minutes. To prevent misconception, the direction of the displacement apparent after a few minutes has been termed the direction of the first decided movement in Table XXXIX., which contains some data as to the direction given by Ellis^[37] and van Bemmelen.^[36] The + sign means an increase, the − sign a decrease of the element. The sign is not invariably the same, it will be understood, but there are in all cases a marked preponderance of changes in the direction shown in the table. The fact that all the stations indicated an increase in horizontal force is of special significance.

Table XXXVI.—Disturbances, and their Annual Distribution.

	Total  Number.	Percentages.
		Winter.	Equinox.	Summer.
Greenwich disturbed days, all, 1848–1902	4,214	33.9	39.2	26.9
Greenwich disturbed days, range 10′ to 30′, 1848–1902	3,830	33.9	39.0	27.1
Greenwich disturbed days range 30′ to 60′, 1848–1902	307	34.5	41.0	24.4
Greenwich disturbed days, range over 60′, 1848–1902	77	29.9	41.6	28.6
Kew highly disturbed days, 1890–1900	209	38.3	41.6	20.1
Greenwich magnetic storms, all, 1848–1903	726	32.1	42.3	25.6
Greenwich magnetic storms, range 20′ to 30′, 1848–1903	392	30.1	43.6	26.3
Greenwich magnetic storms, range over 30′, 1848–1903	334	34.4	40.7	24.9
Greenwich magnetic storms, all, 14 years of S. max.	258	35.3	38.0	26.7
Greenwich magnetic storms, all, 15 years of S. min.	127	28.4	48.0	23.6
Batavia magnetic storms, all, 1883–1899	1,008	32.9	34.9	32.2
Batavia magnetic storms of gradual commencement	679	32.4	34.8	32.8
Batavia magnetic storms of sudden commencement	329	33.7	35.3	31.0

Table XXXVII.—Batavia Magnetic Storms, Diurnal
Distribution (percentages).

Hour.		0	2	4	6	8	10	12	14	16	18	20	22
Beginning ${\Big \{}$	g	5	5	5	6	20	16	7	5	6	9	8	8
	s	7	5	7	10	10	11	10	8	8	9	8	7
Maximum ${\Big \{}$	g	12	10	6	5	4	9	9	6	6	6	12	15
	s	14	7	5	2	2	9	9	5	8	10	13	16
End	all	15	16	19	13	5	3	6	5	4	5	4	5

Table XXXVIII.—Greenwich Magnetic Storms, Diurnal Distribution.

Epoch.		Class.	Total  Number.	Percentages.
				1–8 p.m.	9 p.m.–  4 a.m.	5 a.m.–  noon.
Beginning ${\Bigg \{}$	1848–1903	all	721	60.1	21.9	18.0
	1882–1903	”	276	58.0	18.8	23.2
	1882–1903	sudden	77	45.4	27.3	27.3
End${\Bigg \{}$	1848–1903	all	720	9.4	44.6	46.0
	1882–1903	”	276	7.2	41.7	51.1
	1882–1903	sudden	77	11.7	35.1	53.2

§ 37. That large magnetic disturbances occur simultaneously over large areas was known in the time of Gauss, on whose initiative observations were taken at 5-minute intervals at a number of stations on prearranged term days. During March 1879 and August 1880 some large magnetic storms occurred, and the magnetic curves showing these at a number of stations fitted with Kew pattern magnetographs were compared by W. G. Adams.^[38] He found the more characteristic movements to be, so far as could be judged, simultaneous at all the stations. At comparatively near stations such as Stonyhurst and Kew, or Coimbra and Lisbon, the curves were in general almost duplicates. At Kew and St Petersburg there were usually considerable differences in detail, and the movements were occasionally in opposite directions. The differences between Toronto, Melbourne or Zi-ka-wei and the European stations were still more pronounced. In 1896, on the initiative of M. Eschenhagen,^[39] eye observations of declination and horizontal force were taken at 5-second intervals during prearranged hours at Batavia, Manila, Melbourne and nine European stations. The data from one of these occasions when appreciable disturbance prevailed were published by Eschenhagen, and were subsequently analysed by Ad. Schmidt.^[40] Taking the stations in western Europe, Schmidt drew several series of lines, each series representing the disturbing forces at one instant of time as deduced from the departure of the elements at the several stations from their undisturbed value. The lines answering to any one instant had a general sameness of direction with more or less divergence or convergence, but their general trend varied in a way which suggested to Schmidt the passage of a species of vortex with large but finite velocity.

Table XXXIX.—Direction of First Decided Movement.

Place.	Declination.	Horizontal Force.	Vertical Force.
Pavlovsk	West	+	+
Potsdam	West	+	−
Greenwich	West	+	+
Zi-ka-wei	East	+	−
Kolaba	East	+	−
Batavia	West	+	−
Mauritius	East	+	+
Cape Horn	West	+	−

The conclusion that magnetic disturbances tend to follow one another at nearly equal intervals of time has been reached by several independent observers. J. A. Broun^[41] pronounced for a period of about 26 days, and expressed a belief that a certain zone, or zones, of the sun’s surface might exert a prepotent influence on the earth’s magnetism during several solar rotations. Very similar views were advanced in 1904 by E. W. Maunder,^[35] who was wholly unaware of Broun’s work. Maunder concluded that the period was 27.28 days, coinciding with the sun’s rotation period relative to an observer on the earth. Taking magnetic storms at Greenwich from 1882 to 1903, he found the interval between the commencement of successive storms to approach closely to the above period in a considerably larger number of instances than one would have expected from mere chance. He found several successions of three or four storms, and in one instance of as many as six storms, showing his interval. In a later paper Maunder reached similar results for magnetic storms at Greenwich from 1848 to 1881. Somewhat earlier than Maunder, Arthur Harvey^[42] deduced a period of 27.246 days from a consideration of magnetic disturbances at Toronto. A. Schuster,^[43] examining Maunder’s data mathematically, concluded that they afforded rather strong evidence of a period of about 1/2 (27.28) or 13.6 days. Maunder regarded his results as demonstrating that magnetic disturbances originate in the sun. He regarded the solar action as arising from active areas of limited extent on the sun’s surface, and as propagated along narrow, well defined streams. The active areas he believed to be also the seats of the formation of sun-spots, but believed that their activity might precede and outlive the visible existence of the sun-spot.

Maunder did not discuss the physical nature of the phenomenon, but his views are at least analogous to those propounded somewhat earlier by Svante Arrhenius,^[44] who suggested that small negatively charged particles are driven from the sun by the repulsion of light and reach the earth’s atmosphere, setting up electrical currents, manifest in aurora and magnetic disturbances. Arrhenius’s calculations, for the size of particle which he regarded as most probable, make the time of transmission to the earth slightly under two days. Amongst other theories which ascribe magnetic storms to direct solar action may be mentioned that of Kr. Birkeland,^[45] who believes the vehicle to be cathode rays. Ch. Nordmann^[46] similarly has suggested Röntgen rays. Supposing the sun the ultimate source, it would be easier to discriminate between the theories if the exact time of the originating occurrence could be fixed. For instance, a disturbance that is propagated with the velocity of light may be due to Röntgen rays, but not to Arrhenius’s particles. In support of his theory, Nordmann mentions several cases when conspicuous visual phenomena on the sun have synchronized with magnetic movements on the earth—the best known instance being the apparent coincidence in time of a magnetic disturbance at Kew on the 1st of September 1859 with a remarkable solar outburst seen by R. C. Carrington. Presumably any electrical phenomenon on the sun will set up waves in the aether, so transmission of electric and magnetic disturbances from the sun to the earth with the velocity of light is a certainty rather than a hypothesis; but it by no means follows that the energy thus transmitted can give rise to sensible magnetic disturbances. Also, when considering Nordmann’s coincidences, it must be remembered that magnetic movements are so numerous that it would be singular if no apparent coincidences had been noticed. Another consideration is that the movements shown by ordinary magnetographs are seldom very rapid. During some storms, especially those accompanied by unusually bright and rapidly varying auroral displays, large to and fro movements follow one another in close succession, the changes being sometimes too quick to be registered distinctly on the photographic paper. This, however, is exceptional, even in polar regions where disturbances are largest and most numerous. As a rule, even when the change in the direction of movement in the declination needle seems quite sudden, the movement in one direction usually lasts for several minutes, often for 10, 15 or 30 minutes. Thus the cause to which magnetic disturbances are due seems in many cases to be persistent in one direction for a considerable time.

§ 38. Attempts have been made to discriminate between the theories as to magnetic storms by a critical examination of the phenomena. A general connexion between sun-spot frequency and the amplitude of magnetic movements, regular and irregular, is generally admitted. If it is a case of cause and effect, and the interval between the solar and terrestrial phenomena does not exceed a few hours, then there should be a sensible connexion between corresponding daily values of the sun-spot frequency and the magnetic range. Even if only some sun-spots are effective, we should expect when we select from a series of years two groups of days, the one containing the days of most sun-spots, the other the days of least, that a prominent difference will exist between the mean values of the absolute daily magnetic ranges for the two groups. Conversely, if we take out the days of small and the days of large magnetic range, or the days that are conspicuously quiet and those that are highly disturbed, we should expect a prominent difference between the corresponding mean sun-spot areas. An application of this principle was made by Chree^[19] to the five quiet days a month selected by the astronomer royal between 1890 and 1900. These days are very quiet relative to the average day and possess a much smaller absolute range. One would thus have expected on Birkeland’s or Nordmann’s theory the mean sun-spot frequency derived from Wolfer’s provisional values for these days to be much below his mean value, 41.22, for the eleven years. It proved, however, to be 41.28. This practical identity was as visible in 1892 to 1895, the years of sun-spot maximum, as it was in the years of sun-spot minimum. Use was next made of the Greenwich projected sun-spot areas, which are the result of exact measurement. The days of each month were divided into three groups, the first and third—each normally of ten days—containing respectively the days of largest and the days of least sun-spot area. The mean sun-spot area from group 1 was on the average about five times that for group 3. It was then investigated how the astronomer royal’s quiet days from 1890 to 1900, and how the most disturbed days of the period selected from the Kew^[20] magnetic records, distributed themselves among the three groups of days. Nineteen months were excluded, as containing more than ten days with no sun-spots. The remaining 113 months contained 565 quiet and 191 highly disturbed days, whose distribution was as follows:

	Group 1.	Group 2.	Group 3.
Quiet days	179	195	191
Disturbed days	68	65	58

The group of days of largest sun-spot area thus contained slightly under their share of quiet days and slightly over their share of disturbed days. The differences, however, are not large, and in three years, viz. 1895, 1897 and 1899, the largest number of disturbed days actually occurred in group 3, while in 1895, 1896 and 1899 there were fewer quiet days in group 3 than in group 1. Taking the same distribution of days, the mean value of the absolute daily range of declination at Kew was calculated for the group 1 and the group 3 days of each month. The mean range from the group 1 days was the larger in 57% of the individual months as against 43% in which it was the smaller. When the days of each month were divided into groups according to the absolute declination range at Kew, the mean sun-spot area for the group 1 days (those of largest range) exceeded that for the group 3 days (those of least range) in 55% of the individual months, as against 45% of cases in which it was the smaller.

Taking next the five days of largest and the five days of least range in each month, sun-spot areas were got out not merely for these days themselves, but also for the next subsequent day and the four immediately preceding days in each case. On Arrhenius’s theory we should expect the magnetic range to vary with the sun-spot area, not on the actual day but two days previously. The following figures give the percentage excess or deficiency of the mean sun-spot area for the respective groups of days, relative to the average value for the whole epoch dealt with. n denotes the day to which the magnetic range belongs, n + 1 the day after, n − 1 the day before, and so on. Results are given for 1894 and 1895, the years which were on the whole the most favourable and the least favourable for Arrhenius’s hypothesis, as well as for the whole eleven years.

Table XL.

Day.			n − 4	n − 3	n − 2	n − 1	n	n + 1
Five days of  largest range	${\Bigg \}}$	1894	+12	+ 9	+11	+12	+11	+ 6
		1895	−16	−17	−15	−12	−11	−10
		11 yrs.	+ 9	+ 8	+ 8	+ 7	+ 5	+ 0.5
Five days of   least range	${\Bigg \}}$	1894	−15	−17	−19	−21	−21	−19
		1895	+17	+10	+ 1	− 2	− 2	− 4
		11 yrs.	− 4	− 4	− 7	− 7	− 7	− 6

Taking the 11-year-means we have the sun-spot area practically normal on the day subsequent to the representative day of large magnetic range, but sensibly above its mean on that day and still more so on the four previous days. This suggests an emission from the sun taking a highly variable time to travel to the earth. The 11-year mean data for the five days of least range seem at first sight to point to the same conclusion, but the fact that the deficiency in sun-spot area is practically as prominent on the day after the representative day of small magnetic range as on that day itself, or the previous days, shows that the phenomenon is probably a secondary one. On the whole, taking into account the extraordinary differences between the results from individual years, we seem unable to come to any very positive conclusion, except that in the present state of our knowledge little if any clue is afforded by the extent of the sun’s spotted area on any particular day as to the magnetic conditions on the earth on that or any individual subsequent day. Possibly some more definite information might be extracted by considering the extent of spotted area on different zones of the sun. On theories such as those of Arrhenius or Maunder, effective bombardment of the earth would be more or less confined to spotted areas in the zones nearest the centre of the visible hemisphere, whilst all spots on this hemisphere contribute to the total spotted area. Still the projected area of a spot rapidly diminishes as it approaches the edge of the visible hemisphere, i.e. as it recedes from the most effective position, so that the method employed above gives a preponderating weight to the central zones. One rather noteworthy feature in Table XL. is the tendency to a sequence in the figures in any one row. This seems to be due, at least in large part, to the fact that days of large and days of small sun-spot area tend to occur in groups. The same is true to a certain extent of days of large and days of small magnetic range, but it is unusual for the range to be much above the average for more than 3 or 4 successive days.

§ 39. The records from ordinary magnetographs, even when run at the usual rate and with normal sensitiveness, not infrequently show a repetition of regular or nearly regular small rhythmic movements, lasting sometimes for hours. The amplitude and period on different occasions both vary widely. Periods of 2 to 4 Pulsations. minutes are the most common. W. van Bemmelen^[47] has made a minute examination of these movements from several years’ traces at Batavia, comparing the results with corresponding statistics sent him from Zi-ka-wei and Kew. Table XLI. shows the diurnal variation in the frequency of occurrence of these small movements—called pulsations by van Bemmelen—at these three stations. The Batavia results are from the years 1885 and 1892 to 1898. Of the two sets of data for Zi-ka-wei (i) answers to the years 1897, 1898 and 1900, as given by van Bemmelen, while (ii) answers to the period 1900–1905, as given in the Zi-ka-wei Bulletin for 1905. The Kew data are for 1897. The results are expressed as percentages of the total for the 24 hours. There is a remarkable contrast between Batavia and Zi-ka-wei on the one hand and Kew on the other, pulsations being much more numerous by night than by day at the two former stations, whereas at Kew the exact reverse holds. Van Bemmelen decided that almost all the occasions of pulsation at Zi-ka-wei were also occasions of pulsations at Batavia. The hours of commencement at the two places usually differed a little, occasionally by as much as 20 minutes; but this he ascribed to the fact that the earliest oscillations were too small at one or other of the stations to be visible on the trace. Remarkable coincidence between pulsations at Potsdam and in the north of Norway has been noted by Kr. Birkeland.^[45]

With magnetographs of greater sensitiveness and more open time scales, waves of shorter period become visible. In 1882 F. Kohlrausch^[48] detected waves with a period of about 12 seconds. Eschenhagen^[49] observed a great variety of short period waves, 30 seconds being amongst the most common. Some of the records he obtained suggest the superposition of regular sine waves of different periods. Employing a very sensitive galvanometer to record changes of magnetic induction through a coil traversed by the earth’s lines of force, H. Ebert^[50] has observed vibrations whose periods are but a small fraction of a second. The observations of Kohlrausch and Eschenhagen preceded the recent great development of applications of electrical power, while longer period waves are shown in the Kew curves of 50 years ago, so that the existence of natural waves with periods of from a few seconds up to several minutes can hardly be doubted. Whether the much shorter period waves of Ebert are also natural is more open to doubt, as it is becoming exceedingly difficult in civilized countries to escape artificial disturbances.

Table XLI.—Diurnal Distribution of Pulsations.

Hours.	0–3.	3–6.	6–9.	9–Noon.	Noon–3.	3–6.	6–9.	9–12.
Batavia	28	9	2	6	8	6	13	28
Zi-ka-wei (i)	33	5	2	7	4	4	10	35
Zi-ka-wei (ii)	23	6	8	11	7	5	14	26
Kew	4	8	19	14	22	18	11	4

§ 40. The fact that the moon exerts a small but sensible effect on the earth’s magnetism seems to have been first discovered in 1841 by C. Kreil. Subsequently Sabine^[51] investigated the nature of the lunar diurnal variation in declination Lunar Influence. at Kew, Toronto, Pekin, St Helena, Cape of Good Hope and Hobart. The data in Table XLII. are mostly due to Sabine. They represent the mean lunar diurnal inequality in declination for the whole year. The unit employed is 0′.001, and as in our previous tables + denotes movement to the west. By “mean departure” is meant the arithmetic mean of the 24 hourly departures from the mean value for the lunar day; the range is the difference between the algebraically greatest and least of the hourly values. Not infrequently the mean departure gives the better idea of the importance of an inequality, especially when as in the present case two maxima and minima occur in the day. This double daily period is unusually prominent in the case of the lunar diurnal inequality, and is seen in the other elements as well as in the declination.

Table XLII.—Lunar Diurnal Inequality of Declination (unit 0′.001).

Lunar Hour.	Kew. 1858–1862.	Toronto. 1843–1848.	Batavia. 1883–1899.	St Helena. 1843–1847.	Cape. 1842–1846.	Hobart. 1841–1848.
0	+103	+315	−70	− 43	−148	− 98
1	+160	+275	−63	− 5	−107	−138
2	+140	+158	−39	+ 37	− 35	−142
3	+ 33	+ 2	− 8	+ 70	+ 43	−107
4	+ 10	−153	+38	+ 85	+108	− 45
5	− 67	−265	+63	+ 77	+140	+ 27
6	−150	−302	+87	+ 48	+132	+ 88
7	−188	−255	+77	+  5	+ 82	+122
8	−160	−137	+40	− 43	+  5	+120
9	− 78	+  7	− 4	− 82	− 78	+ 82
10	+  2	+178	−45	−102	−143	+ 17
11	+ 92	+288	−80	− 98	−177	− 57
12	+160	+323	−87	− 73	−165	−120
13	+188	+272	−68	− 32	−112	−152
14	+158	+148	−43	+ 13	− 30	−147
15	+ 90	− 17	− 8	+ 52	+ 58	−105
16	+ 10	−180	+30	+ 73	+132	− 35
17	− 85	−297	+62	+ 73	+172	+ 45
18	−142	−337	+72	+ 52	+168	+112
19	−163	−290	+68	+ 17	+122	+152
20	−147	−170	+52	− 25	+ 45	+152
21	−123	−   7	+ 8	− 58	− 40	+113
22	− 40	+155	−28	− 73	−112	+ 47
23	+ 27	+265	−56	− 68	−153	− 30
Mean De- parture ${\Big \}}$	105	200	50	54	104	93
Range	376	660	174	187	349	304

Lunar action has been specially studied in connexion with observations from India and Java. Broun^[52] at Trivandrum and C. Chambers^[53] at Kolaba investigated lunar action from a variety of aspects. At Batavia van der Stok^[54] and more recently S. Figee^[55] have carried out investigations involving an enormous amount of computation. Table XLIII. gives a summary of Figee’s results for the mean lunar diurnal inequality at Batavia, for the two half-yearly periods April to September (Winter or W.), and October to March (S.). The + sign denotes movement to the west in the case of declination, but numerical increase in the case of the other elements. In the case of H and T (total force) the results for the two seasons present comparatively small differences, but in the case of D, I and V the amplitude and phase both differ widely. Consequently a mean lunar diurnal variation derived from all the months of the year gives at Batavia, and presumably at other tropical stations, an inadequate idea of the importance of the lunar influence. In January Figee finds for the range of the lunar diurnal inequality 0′.62 in D, 3.1γ in H and 3.5γ in V, whereas the corresponding ranges in June are only 0′.13, 1.1γ and 2.2γ respectively. The difference between summer and winter is essentially due to solar action, thus the lunar influence on terrestrial magnetism is clearly a somewhat complex phenomenon. From a study of Trivandrum data, Broun concluded that the action of the moon is largely dependent on the solar hour at the time, being on the average about twice as great for a day hour as for a night hour. Figee’s investigations at Batavia point to a similar conclusion. Following a method suggested by Van der Stok, Figee arrives at a numerical estimate of the “lunar activity” for each hour of the solar day, expressed in terms of that at noon taken as 100. In summer, for instance, in the case of D he finds the “activity” varying from 114 at 10 a.m. to only 8 at 9 p.m.; the corresponding extremes in the case of H are 139 at 10 a.m. and 54 at 6 a.m.

Table XLIII.—Lunar Diurnal Inequality at Batavia in Winter and Summer.

	Declination  (unit 0′.001).		Inclination, S.  (unit 0′.001).		H.  (unit 0.01γ).		V.  (unit 0.01γ).		T. (unit0.01γ).
Lunar Hour.	W.	S.	W.	S.	W.	S.	W.	S.	W.	S.
0	+30	−170	− 1	+25	−15	− 56	− 9	+ 4	− 17	−47
1	+21	−147	−23	+49	−40	− 87	−54	+20	− 61	−67
2	+ 5	− 83	−49	+69	−25	−107	−82	+37	− 62	−76
3	− 5	− 12	−51	+47	−21	− 76	−83	+24	− 59	−55
4	+ 1	+ 76	−37	+43	−13	− 59	−58	+18	− 39	−38
5	− 8	+134	−23	+12	+10	− 9	−27	+11	− 4	− 3
6	− 7	+181	− 2	−21	+21	+ 43	+ 9	− 6	+ 23	+35
7	−10	+164	+30	−12	+23	+ 45	+55	+ 8	+ 47	+43
8	− 7	+ 86	+36	−21	+38	+ 52	+71	− 1	+ 68	+45
9	− 8	0	+28	−23	+46	+ 30	+64	−16	+ 71	+19
10	− 5	− 85	+34	−20	+13	+ 13	+54	−21	+ 38	+ 1
11	−15	−144	+27	−11	−12	− 6	+31	−19	+ 5	−15
12	− 9	−164	+19	− 5	−47	− 23	0	−19	− 41	−29
13	+ 1	−136	− 3	+17	−59	− 46	−36	− 2	− 69	−41
14	− 7	− 79	−13	+27	−66	− 44	−55	+14	− 84	−32
15	− 8	−  8	−32	+25	−53	− 37	−74	+14	− 82	−26
16	−12	+ 72	−37	+25	−34	− 17	−70	+26	− 64	− 2
17	−13	+137	−33	+ 4	− 1	+ 28	−47	+21	− 24	+35
18	−21	+165	− 2	−10	+20	+ 47	+ 8	+12	+ 21	+47
19	−12	+147	+21	−42	+44	+ 81	+53	−14	+ 64	+64
20	+10	+ 95	+21	−62	+75	+107	+71	−28	+100	+80
21	+13	+  4	+26	−70	+65	+ 98	+72	−44	+ 92	+65
22	+25	− 82	+35	−41	+35	+ 35	+68	−38	+ 64	+12
23	+36	−147	+34	− 4	− 7	− 14	+44	−13	+ 15	−19
Mean De- parture ${\Big \}}$	12	150	26	29	33	48	50	18	51	37
Range	57	351	87	139	141	214	155	81	184	156

The question whether lunar influence increases with sun-spot frequency is obviously of considerable theoretical interest. Balfour Stewart in the 9th edition of this encyclopaedia gave some data indicating an appreciably enhanced lunar influence at Trivandrum during years of sun-spot maximum, but he hesitated to accept the result as finally proved. Figee recently investigated this point at Batavia, but with inconclusive results. Attempts have also been made to ascertain how lunar influence depends on the moon’s declination and phase, and on her distance from the earth. The difficulty in these investigations is that we are dealing with a small effect, and a very long series of data would be required satisfactorily to eliminate other periodic influences.

§ 41. From an analysis of seventeen years data at St Petersburg and Pavlovsk, Leyst^[56] concluded that all the principal planets sensibly influence the earth’s magnetism. According to his figures, all the planets except Mercury—whose influence he found opposite to that of the others—when Planetary Influence. nearest the earth tended to deflect the declination magnet at St Petersburg to the west, and also increased the range of the diurnal inequality of declination, the latter effect being the more conspicuous. Schuster,^[57] who has considered the evidence advanced by Leyst from the mathematical standpoint, considers it to be inconclusive.

§ 42. The best way of carrying out a magnetic survey depends on where it has to be made and on the object in view. The object that probably still comes first in importance is a knowledge of the declination, of sufficient accuracy for navigation in all navigable waters. One might thus infer that Magnetic Surveys. magnetic surveys consist mainly of observations at sea. This cannot however be said to be true of the past, whatever it may be of the future, and this for several reasons. Observations at sea entail the use of a ship, specially constructed so as to be free from disturbing influence, and so are inherently costly; they are also apt to be of inferior accuracy. It might be possible in quiet weather, in a large vessel free from vibration, to observe with instruments of the highest precision such as a unifilar magnetometer, but in the ordinary surveying ship apparatus of less sensitiveness has to be employed. The declination is usually determined with some form of compass. The other elements most usually found directly at sea are the inclination and the total force, the instrument employed being a special form of inclinometer, such as the Fox circle, which was largely used by Ross in the Antarctic, or in recent years the Lloyd-Creak. This latter instrument differs from the ordinary dip-circle fitted for total force observations after H. Lloyd’s method mainly in that the needles rest in pivots instead of on agate edges. To overcome friction a projecting pin on the framework is scratched with a roughened ivory plate.

The most notable recent example of observations at sea is afforded by the cruises of the surveying ships “Galilee” and “Carnegie” under the auspices of the Carnegie Institution of Washington, which includes in its magnetic programme a general survey. To see where the ordinary land survey assists navigation, let us take the case of a country with a long seaboard. If observations were taken every few miles along the coast results might be obtained adequate for the ordinary wants of coasting steamers, but it would be difficult to infer what the declination would be 50 or even 20 miles off shore at any particular place. If, however, the land area itself is carefully surveyed, one knows the trend of the lines of equal declination, and can usually extend them with considerable accuracy many miles out to sea. One also can tell what places if any on the coast suffer from local disturbances, and thus decide on the necessity of special observations. This is by no means the only immediately useful purpose which is or may be served by magnetic surveys on land. In Scandinavia use has been made of magnetic observations in prospecting for iron ore. There are also various geological and geodetic problems to whose solution magnetic surveys may afford valuable guidance. Among the most important recent surveys may be mentioned those of the British Isles by A. Rücker and T. E. Thorpe,^[58] of France and Algeria by Moureaux,^[59] of Italy by Chistoni and Palazzo,^[60] of the Netherlands by Van Ryckevorsel,^[61] of South Sweden by Carlheim Gyllenskiöld,^[62] of Austria-Hungary by Liznar,^[63] of Japan by Tanakadate,^[64] of the East Indies by Van Bemmelen, and South Africa by J. C. Beattie. A survey of the United States has been proceeding for a good many years, and many results have appeared in the publications of the U.S. Coast and Geodetic Survey, especially Bauer’s Magnetic Tables and Magnetic Charts, 1908. Additions to our knowledge may also be expected from surveys of India, Egypt and New Zealand.

For the satisfactory execution of a land survey, the observers must have absolute instruments such as the unifilar magnetometer and dip circle, suitable for the accurate determination of the magnetic elements, and they must be able to fix the exact positions of the spots where observations are taken. If, as usual, the survey occupies several years, what is wanted is the value of the elements not at the actual time of observation, but at some fixed epoch, possibly some years earlier or later. At a magnetic observatory, with standardized records, the difference between the values of a magnetic element at any two specified instants can be derived from the magnetic curves. But at an ordinary survey station, at a distance from an observatory, the information is not immediately available. Ordinarily the reduction to a fixed epoch is done in at least two stages, a correction being applied for secular change, and a second for the departure from the mean value for the day due to the regular diurnal inequality and to disturbance.

The reduction to a fixed epoch is at once more easy and more accurate if the area surveyed contains, or has close to its borders, a well distributed series of magnetic observatories, whose records are comparable and trustworthy. Throughout an area of the size of France or Germany, the secular change between any two specified dates can ordinarily be expressed with sufficient accuracy by a formula of the type

δ = δ₀ + a(l − l₀) + b(λ − λ₀)  .   .  (i),

where δ denotes secular change, l latitude and λ longitude, the letters with suffix 0 relating to some convenient central position. The constants δ₀, a, b are to be determined from the observed secular changes at the fixed observatories whose geographical co-ordinates are accurately known. Unfortunately, as a rule, fixed observatories are few in number and not well distributed for survey purposes; thus the secular change over part at least of the area has usually to be found by repeating the observations after some years at several of the field stations. The success attending this depends on the exactitude with which the sites can be recovered, on the accuracy of the observations, and on the success with which allowance is made for diurnal changes, regular and irregular. It is thus desirable that the observations at repeat stations should be taken at hours when the regular diurnal changes are slow, and that they should not be accepted unless taken on days that prove to be magnetically quiet. Unless the secular change is exceptionally rapid, it will usually be most convenient in practice to calculate it from or to the middle of the month, and then to allow for the difference between the mean value for the month and the value at the actual hour of observation. There is here a difficulty, inasmuch as the latter part of the correction depends on the diurnal inequality, and so on the local time of the station. No altogether satisfactory method of surmounting this difficulty has yet been proposed. Rücker and Thorpe in their British survey assumed that the divergence from the mean value at any hour at any station might be regarded as made up of a regular diurnal inequality, identical with that at Kew when both were referred to local time, and of a disturbance element identical with that existing at the same absolute time at Kew. Suppose, for instance, that at hour h G.M.T. the departure at Kew from the mean value for the month is d, then the corresponding departure from the mean at a station λ degrees west of Kew is d − e, where e is the increase in the element at Kew due to the regular diurnal inequality between hour h − λ/15 and hour h. This procedure is simple, but is exposed to various criticisms. If we define a diurnal inequality as the result obtained by combining hourly readings from all the days of a month, we can assign a definite meaning to the diurnal inequality for a particular month of a particular year, and after the curves have been measured we can give exact numerical figures answering to this definition. But the diurnal inequality thus obtained differs, as has been pointed out, from that derived from a limited number of the quietest days of the month, not merely in amplitude but in phase, and the view that the diurnal changes on any individual day can be regarded as made up of a regular diurnal inequality of definite character and of a disturbance element is an hypothesis which is likely at times to be considerably wide of the mark. The extent of the error involved in assuming the regular diurnal inequality the same in the north of Scotland, or the west of Ireland, as in the south-east of England remains to be ascertained. As to the disturbance element, even if the disturbing force were of given magnitude and direction all over the British Isles—which we now know is often very far from the case—its effects would necessarily vary very sensibly owing to the considerable variation in the direction and intensity of the local undisturbed force. If observations were confined to hours at which the regular diurnal changes are slow, and only those taken on days of little or no disturbance were utilized, corrections combining the effects of regular and irregular diurnal changes could be derived from the records of fixed observations, supposed suitably situated, combined in formulae of the same type as (i).

§ 43. The field results having been reduced to a fixed epoch, it remains to combine them in ways likely to be useful. In most cases the results are embodied in charts, usually of at least two kinds, one set showing only general features, the other the chief local peculiarities. Charts of the first kind resemble the world charts (figs. 1 to 4) in being free from sharp twistings and convolutions. In these the declination for instance at a fixed geographical position on a particular isogonal is to be regarded as really a mean from a considerable surrounding area.

Various ways have been utilized for arriving at these terrestrial isomagnetics—as Rücker and Thorpe call them—of which an elaborate discussion has been made by E. Mathias.^[65] From a theoretical standpoint the simplest method is perhaps that employed by Liznar for Austria-Hungary. Let l and λ represent latitude and longitude relative to a certain central station in the area. Then assume that throughout the area the value E of any particular magnetic element is given by a formula

E = E₀ + al + bλ + cl ² + dλ² + elλ,

where E₀, a, b, c, d, e are absolute constants to be determined from the observations. When determining the constants, we write for E in the equation the observed value of the element (corrected for secular change, &c.) at each station, and for l and λ the latitude and longitude of the station relative to the central station. Thus each station contributes an equation to assist in determining the six constants. They can thus be found by least squares or some simpler method. In Liznar’s case there were 195 stations, so that the labour of applying least squares would be considerable. This is one objection to the method. A second is that it may allow undesirably large weight to a few highly disturbed stations. In the case of the British Isles, Rücker and Thorpe employed a different method. The area was split up into districts. For each district a mean was formed of the observed values of each element, and the mean was assigned to an imaginary central station, whose geographical co-ordinates represented the mean of the geographical co-ordinates of the actual stations. Want of uniformity in the distribution of the stations may be allowed for by weighting the results. Supposing E₀ the value of the element found for the central station of a district, it was assumed that the value E at any actual station whose latitude and longitude exceeded those of the central station by l and λ was given by E = E₀ + al + bλ, with a and b constants throughout the district. Having found E₀, a and b, Rücker and Thorpe calculated values of the element for points defined by whole degrees of longitude (from Greenwich) and half degrees of latitude. Near the common border of two districts there would be two calculated values, of which the arithmetic mean was accepted.

The next step was to determine by interpolation where isogonals—or other isomagnetic lines—cut successive lines of latitude. The curves formed by joining these successive points of intersection were called district lines or curves. Rücker and Thorpe’s next step was to obtain formulae by trial, giving smooth curves of continuous curvature—terrestrial isomagnetics—approximating as closely as possible to the district lines. The curves thus obtained had somewhat complicated formulae. For instance, the isogonals south of 54°.5 latitude were given for the epoch Jan. 1, 1891 by

D = 18° 37′ + 18′.5 (l − 49.5) − 3′.5 cos {45° (l − 49.5) }

+ {26′.3 + 1′.5 (l − 49.5) } (λ − 4) + 0′.01 (λ − 4)² (l − 54.5)²,

where D denotes the westerly declination. Supposing, what is at least approximately true, that the secular change in Great Britain since 1891 has been uniform south of lat. 54°.5, corresponding formulae for the epochs Jan. 1, 1901, and Jan. 1, 1906, could be obtained by substituting for 18° 37′ the values 17° 44′ and 17° 24′ respectively. In their very laborious and important memoir E. Mathias and B. Baillaud^[65] have applied to Rücker and Thorpe’s observations a method which is a combination of Rücker and Thorpe’s and of Liznar’s. Taking Rücker and Thorpe’s nine districts, and the magnetic data found for the nine imaginary central stations, they employed these to determine the six constants of Liznar’s formula. This is an immense simplification in arithmetic. The declination formula thus obtained for the epoch Jan. 1, 1891, was

D = 20° 45′.89 + .53474λ + .34716l + .000021λ²

+ .000343lλ − .000239l ²,

where l + (53° 30′.5) represents the latitude, and (λ + 5° 35′.2) the west longitude of the station. From this and the corresponding formulae for the other elements, values were calculated for each of Rücker and Thorpe’s 882 stations, and these were compared with the observed values. A complete record is given of the differences between the observed and calculated values, and of the corresponding differences obtained by Rücker and Thorpe from their own formulae. The mean numerical (calculated ~ observed) differences from the two different methods are almost exactly the same—being approximately 10′ for declination, 5′1/2 for inclination, and 70γ for horizontal force. The applications by Mathias^[65] of his method to the survey data of France obtained by Moureaux, and those of the Netherlands obtained by van Rïjckevorsel, appear equally successful. The method dispenses entirely with district curves, and the parabolic formulae are perfectly straightforward both to calculate and to apply; they thus appear to possess marked advantages. Whether the method could be applied equally satisfactorily to an area of the size of India or the United States actual trial alone would show.

§ 44. Rücker and Thorpe regarded their terrestrial isomagnetics and the corresponding formulae as representing the normal field that would exist in the absence of disturbances peculiar to the neighbourhood. Subtracting the forces derived from the formulae from those observed, weLocal Disturbances. obtain forces which may be ascribed to regional disturbance.

When the vertical disturbing force is downwards, or the observed vertical component larger than the calculated, Rücker and Thorpe regard it as positive, and the loci where the largest positive values occur they termed ridge lines. The corresponding loci where the largest negative values occur were called valley lines. In the British Isles Rücker and Thorpe found that almost without exception, in the neighbourhood of a ridge line, the horizontal component of the disturbing force pointed towards it, throughout a considerable area on both sides. The phenomena are similar to what would occur if ridge lines indicated the position of the summits of underground masses of magnetic material, magnetized so as to attract the north-seeking pole of a magnet. Rücker and Thorpe were inclined to believe in the real existence of these subterranean magnetic mountains, and inferred that they must be of considerable extent, as theory and observation alike indicate that thin basaltic sheets or dykes, or limited masses of trap rock, produce no measurable magnetic effect except in their immediate vicinity. In support of their conclusions, Rücker and Thorpe dwell on the fact that in the United Kingdom large masses of basalt such as occur in Skye, Mull, Antrim, North Wales or the Scottish coalfield, are according to their survey invariably centres of attraction for the north-seeking pole of a magnet. Various cases of repulsion have, however, been described by other observers in the northern hemisphere.

§ 45. Rücker and Thorpe did not make a very minute examination of disturbed areas, so that purely local disturbances larger than any noticed by them may exist in the United Kingdom. But any that exist are unlikely to rival some that have been observed elsewhere, notably those in the province of Kursk in Russia described by Moureaux^[66] and by E. Leyst.^[67] In Kursk Leyst observed declinations varying from 0° to 360°, inclinations varying from 39°.1 to 90°; he obtained values of the horizontal force varying from 0 to 0.856 C.G.S., and values of the vertical force varying from 0.371 to 1.836. Another highly disturbed Russian district Krivoi Rog (48° N. lat. 33° E. long.) was elaborately surveyed by Paul Passalsky.^[68] The extreme values observed by him differed, the declination by 282° 40′, the inclination by 41° 53′, horizontal force by 0.658, and vertical force by 1.358. At one spot a difference of 116°1/2 was observed between the declinations at two positions only 42 metres apart. In cases such as the last mentioned, the source of disturbance comes presumably very near the surface. It is improbable that any such enormously rapid changes of declination can be experienced anywhere at the surface of a deep ocean. But in shallow water disturbances of a not very inferior order of magnitude have been met with. Possibly the most outstanding case known is that of an area, about 3 m. long by 11/4 m. at its widest, near Port Walcott, off the N.W. Australian coast. The results of a minute survey made here by H.M.S. “Penguin” have been discussed by Captain E. W. Creak.^[69] Within the narrow area specified, declination varied from 26° W. to 56° E., and inclination from 50° to nearly 80°, the observations being taken some 80 ft. above sea bottom. Another noteworthy case, though hardly comparable with the above, is that of East Loch Roag at Lewis in the Hebrides. A survey by H.M.S. “Research” in water about 100 ft. deep—discussed by Admiral A. M. Field^[70]—showed a range of 11° in declination. The largest observed disturbances in horizontal and vertical force were of the order 0.02 and 0.05 C.G.S. respectively. An interesting feature in this case was that vertical force was reduced, there being a well-marked valley line.

In some instances regional magnetic disturbances have been found to be associated with geodetic anomalies. This is true of an elongated area including Moscow, where observations were taken by Fritsche.^[71] Again, Eschenhagen^[72] detected magnetic anomalies in an area including the Harz Mountains in Germany, where deflections of the plumb line from the normal had been observed. He found a magnetic ridge line running approximately parallel to the line of no deflection of the plumb line.

§ 46. A question of interest, about which however not very much is known, is the effect of local disturbance on secular change and on the diurnal inequality. The determination of secular change in a highly disturbed locality is difficult, because an unintentional slight change in the spot where the observations are made may wholly falsify the conclusions drawn. When the disturbed area is very limited in extent, the magnetic field may reasonably be regarded as composed of the normal field that would have existed in the absence of local disturbance, plus a disturbance field arising from magnetic material which approaches nearly if not quite to the surface. Even if no sensible change takes place in the disturbance field, one would hardly expect the secular change to be wholly normal. The changes in the rectangular components of the force may possibly be the same as at a neighbouring undisturbed station, but this will not give the same change in declination and inclination. In the case of the diurnal inequality, the presumption is that at least the declination and inclination changes will be influenced by local disturbance. If, for example, we suppose the diurnal inequality to be due to the direct influence of electric currents in the upper atmosphere, the declination change will represent the action of the component of a force of given magnitude which is perpendicular to the position of the compass needle. But when local disturbance exists, the direction of the needle and the intensity of the controlling field are both altered by the local disturbance, so it would appear natural for the declination changes to be influenced also. This conclusion seems borne out by observations made by Passalsky^[68] at Krivoi Rog, which showed diurnal inequalities differing notably from those experienced at the same time at Odessa, the nearest magnetic observatory. One station where the horizontal force was abnormally low gave a diurnal range of declination four times that at Odessa; on the other hand, the range of the horizontal force was apparently reduced. It would be unsafe to draw general conclusions from observations at two or three stations, and much completer information is wanted, but it is obviously desirable to avoid local disturbance when selecting a site for a magnetic observatory, assuming one’s object is to obtain data reasonably applicable to a large area. In the case of the older observatories this consideration seems sometimes to have been lost sight of. At Mauritius, for instance, inside of a circle of only 56 ft. radius, having for centre the declination pillar of the absolute magnetic hut of the Royal Alfred Observatory, T. F. Claxton^[73] found that the declination varied from 4° 56′ to 13° 45′ W., the inclination from 50° 21′ to 58° 34′ S., and the horizontal force from 0.197 to 0.244 C.G.S. At one spot he found an alteration of 1°1/3 in the declination when the magnet was lowered from 4 ft. above the ground to 2. Disturbances of this order could hardly escape even a rough investigation of the site.

§ 47. If we assume the magnetic force on the earth’s surface derivable from a potential V, we can express V as the sum of two series of solid spherical harmonics, one containing negative, the other positive integral powers of the radius vector r from the earth’s centre. Let λ denote east longitude from Greenwich, and let Gaussian Potential and Constants. μ = cos (1/2π − l), where l is latitude; and also let

Hmn = (1 − µ2)1/2m ${\Big [}$ µn−m −	(n − m) (n − m − 1)	µn−m−2 + . . . ${\Big ]}$,
	2 (2n − 1)

where n and m denote any positive integers, m being not greater than n. Then denoting the earth’s radius by R, we have

V / R = Σ (R / r)ⁿ⁺¹ [Hmn (g mn cos mλ + hmn sin mλ) ]

+ Σ (r / R)ⁿ [Hmn (g m−n cos mλ + hm−n sin mλ) ],

where Σ denotes summation of m from 0 to n, followed by summation of n from 0 to ∞. In this equation g mn, &c. are constants, those with positive suffixes being what are generally termed Gaussian constants. The series with negative powers of r answers to forces with a source internal to the earth, the series with positive powers to forces with an external source. Gauss found that forces of the latter class, if existent, were very small, and they are usually left out of account. There are three Gaussian constants of the first order, g₁0, g₁¹, h₁¹, five of the second order, seven of the third, and so on. The coefficient of a Gaussian constant of the nth order is a spherical harmonic of the nth degree. If R be taken as unit length, as is not infrequent, the first order terms are given by

V₁ = r ⁻² [g₁0 sin l + (g₁¹ cos λ + h₁¹ sin λ) cos l].

The earth is in reality a spheroid, and in his elaborate work on the subject J. C. Adams^[74] develops the treatment appropriate to this case. Here we shall as usual treat it as spherical. We then have for the components of the force at the surface

X = −R⁻¹ (1 − µ²)^1/2 (dV / dµ) towards the astronomical north,

Y = −R⁻¹ (1 − µ²)^−1/2 (dV / dλ) towards the astronomical west,

Z = −dV / dr vertically downwards.

Supposing the Gaussian constants known, the above formulae would give the force all over the earth’s surface. To determine the Gaussian constants we proceed of course in the reverse direction, equating the observed values of the force components to the theoretical values involving g mn, &c. If we knew the values of the component forces at regularly distributed stations all over the earth’s surface, we could determine each Gaussian constant independently of the others. Our knowledge however of large regions, especially in the Arctic and Antarctic, is very scanty, and in practice recourse is had to methods in which the constants are not determined independently. The consequence is unfortunately that the values found for some of the constants, even amongst the lower orders, depend very sensibly on how large a portion of the polar regions is omitted from the calculations, and on the number of the constants of the higher orders which are retained.

Table XLIV.—Gaussian Constants of the First Order.

	1829 Erman- Petersen.	1830 Gauss.	1845 Adams.	1880 Adams.	1885  Neumayer.	1885 Schmidt.	1885 Fritsche.
g10	+.32007	+.32348	+.32187	+.31684	+.31572	+.31735	+.31635
g11	+.02835	+.03111	+.02778	+.02427	+.02481	+.02356	+.02414
h11	−.06011	−.06246	−.05783	−.06030	−.06026	−.05984	−.05914

Table XLIV. gives the values obtained for the Gaussian constants of the first order in some of the best-known computations, as collected by W. G. Adams.^[75]

§ 48. Allowance must be made for the difference in the epochs, and for the fact that the number of constants assumed to be worth retaining was different in each case. Gauss, for instance, assumed 24 constants sufficient, whilst in obtaining the results given in the table J. C. Adams retained 48. Some idea of the uncertainty thus arising may be derived from the fact that when Adams assumed 24 constants sufficient, he got instead of the values in the table the following:—

	g10	g11	h11
1842–1845	+.32173	+.02833	−.05820
1880	+.31611	+.02470	−.06071

Some of the higher constants were relatively much more affected. Thus, on the hypotheses of 48 and of 24 constants respectively, the values obtained for g₂0 in 1842–1845 were -.00127 and -.00057, and those obtained for h₃¹ in 1880 were +.00748 and +.00573. It must also be remembered that these values assume that the series in positive powers of r, with coefficients having negative suffixes, is absolutely non-existent. If this be not assumed, then in any equation determing X or Y, g mn must be replaced by g mn + g m−n, and in any equation determining Z by g mn − {n/(n + 1)} g mn; similar remarks apply to hmn and hm−n. It is thus theoretically possible to check the truth of the assumption that the positive power series is non-existent by comparing the values obtained for g mn and hmn from the X and Y or from the Z equations, when g m−n and hm−n are assumed zero. If the values so found differ, values can be found for g m−n and hm−n which will harmonize the two sets of equations. Adams gives the values obtained from the X, Y and the Z equations separately for the Gaussian constants. The following are examples of the values thence deducible for the coefficients of the positive power series:—

	g−10	g−11	h−11	g−40	g−50	g−60
1842–1845	+.0018	−.0002	−.0014	+.0064	+.0072	+.0124
1880	−.0002	−.0012	+.0015	−.0043	−.0021	−.0013

Compared to g₄0, g₅0 and g₆0 the values here found for g₋₄0, g₋₅0 and g₋₆0 are far from insignificant, and there would be no excuse for neglecting them if the observational data were sufficient and reliable. But two outstanding features claim attention, first the smallness of g₋₁0, g₋₁¹ and h₋₁¹, the coefficients least likely to be affected by observational deficiencies, and secondly the striking dissimilarity between the values obtained for the two epochs. The conclusion to which these and other facts point is that observational deficiencies, even up to the present date, are such that no certain conclusion can be drawn as to the existence or non-existence of the positive power series. It is also to be feared that considerable uncertainties enter into the values of most of the Gaussian constants, at least those of the higher orders. The introduction of the positive power series necessarily improves the agreement between observed and calculated values of the force, but it is more likely than not to be disadvantageous physically, if the differences between observed values and those calculated from the negative power series alone arise in large measure from observational deficiencies.

Table XLV.—Axis and Moment of First Order Gaussian Coefficients.

Epoch.	Authority for Constants.	North Latitude.	West  Longitude.	M/R3 in  G.C.S. units.
		°   ′	°   ′
1650	H. Fritsche	82   50	42   55	.3260
1836	”	78   27	63   35	.3262
1845	J. C. Adams	78   44	64   20	.3282
1880	”	78   24	68    4	.3234
1885	Neumayer-Petersen and Bauer	78    3	67    3	.3224
1885	Neumayer, Schmidt	78   34	68   31	.3230

§ 49. The first order Gaussian constants have a simple physical meaning. The terms containing them represent the potential arising from the uniform magnetization of a sphere parallel to a fixed axis, the moment M of the spherical magnet being given by

M = R³ { (g₁0)² + (g₁¹)² + (h₁¹)²}^1/2,

where R is the earth’s radius. The position of the north end of the axis of this uniform magnetization and the values of M/R³, derived from the more important determinations of the Gaussian constants, are given in Table XLV. The data for 1650 are of somewhat doubtful value. If they were as reliable as the others, one would feel greater confidence in the reality of the apparent movement of the north end of the axis from east to west. The table also suggests a slight diminution in M since 1845, but it is open to doubt whether the apparent change exceeds the probable error in the calculated values. It should be carefully noticed that the data in the table apply only to the first order Gaussian terms, and so only to a portion of the earth’s magnetization, and that the Gaussian constants have been calculated on the assumption that the negative power series alone exists. The field answering to the first order terms—or what Bauer has called the normal field—constitutes much the most important part of the whole magnetization. Still what remains is very far from negligible, save for rough calculations. It is in fact one of the weak points in the Gaussian analysis that when one wishes to represent the observed facts with high accuracy one is obliged to retain so many terms that calculation becomes burdensome.

§ 50. The possible existence of a positive power series is not the only theoretical uncertainty in the Gaussian analysis. There is the further possibility that part of the earth’s magnetic field may not answer to a potential at all. Schmidt^[76] Earth-air Currents. in his calculation of Gaussian constants regarded this as a possible contingency, and the results he reached implied that as much as 2 or 3% of the entire field had no potential. If the magnetic force F on the earth’s surface comes from a potential, then the line integral ∫F ds taken round any closed circuit s should vanish. If the integral does not vanish, it equals 4πI, where I is the total electric current traversing the area bounded by s. A + sign in the result of the integration means that the current is downwards (i.e. from air to earth) or upwards, according as the direction of integration round the circuit, as viewed by an observer above ground, has been clockwise or anti-clockwise. In applications of the formula by W. von Bezold^[77] and Bauer^[78] the integral has been taken along parallels of latitude in the direction west to east. In this case a + sign indicates a resultant upward current over the area between the parallel of latitude traversed and the north geographical pole. The difference between the results of integration round two parallels of latitude gives the total vertical current over the zone between them. Schmidt’s final estimate of the average intensity of the earth-air current, irrespective of sign, for the epoch 1885 was 0.17 ampere per square kilometre. Bauer employing the same observational data as Schmidt, reached somewhat similar conclusions from the differences between integrals taken round parallels of latitude at 5° intervals from 60° N. to 60° S. H. Fritsche^[79] treating the problem similarly, but for two epochs, 1842 and 1885, got conspicuously different results for the two epochs, Bauer^[80] has more recently repeated his calculations, and for three epochs, 1842–1845 (Sabine’s charts), 1880 (Creak’s charts), and 1885 (Neumayer’s charts), obtaining the mean value of the current per sq. km. for 5° zones. Table XLVI. is based on Bauer’s figures, the unit being 0.001 ampere, and + denoting an upwardly directed current.

Table XLVI.—Earth-air Currents, after Bauer.

Latitude.	Northern Hemisphere.			Southern Hemisphere.
	1842–5.	1880.	1885.	1842–5.	1880.	1885.
0° to 15°	− 1	−32	−34	+66	+ 30	+ 36
15° to 30°	−70	−59	−68	+ 2	− 62	− 63
30° to 45°	+ 3	+14	−22	+26	− 11	− 14
45° to 60°	−31	−21	+78	+ 5	+276	+213

In considering the significance of the data in Table XLVI., it should be remembered that the currents must be regarded as mean values derived from all hours of the day, and all months of the year. Currents which were upwards during certain hours of the day, and downwards during others, would affect the diurnal inequality; while currents which were upwards during certain months, and downwards during others, would cause an annual inequality in the absolute values. Thus, if the figures be accepted as real, we must suppose that between 15° N. and 30° N. there are preponderatingly downward currents, and between 0° S. and 15° S. preponderatingly upward currents. Such currents might arise from meteorological conditions characteristic of particular latitudes, or be due to the relative distribution of land and sea; but, whatever their cause, any considerable real change in their values between 1842 and 1885 seems very improbable. The most natural cause to which to attribute the difference between the results for different epochs in Table XLVI. is unquestionably observational deficiencies. Bauer himself regards the results for latitudes higher than 45° as very uncertain, but he seems inclined to accept the reality of currents of the average intensity of 1/30 ampere per sq. km. between 45° N. and 45° S.

Currents of the size originally deduced by Schmidt, or even those of Bauer’s latest calculations, seem difficult to reconcile with the results of atmospheric electricity (q.v.).

§ 51. There is no single parallel of latitude along the whole of which magnetic elements are known with high precision. Thus results of greater certainty might be hoped for from the application of the line integral to well surveyed countries. Such applications have been made, e.g. to Great Britain by Rücker,^[81] and to Austria by Liznar,^[82] but with negative results. The question has also been considered in detail by Tanakadate^[64] in discussing the magnetic survey of Japan. He makes the criticism that the taking of a line integral round the boundary of a surveyed area amounts to utilizing the values of the magnetic elements where least accurately known, and he thus considers it preferable to replace the line integral by the surface integral.

4πI = ∬(dY/dx − dX/dy) dxdy.

He applied this formula not merely to his own data for Japan, but also to British and Austrian data of Rücker and Thorpe and of Liznar. The values he ascribes to X and Y are those given by the formulae calculated to fit the observations. The result reached was “a line of no current through the middle of the country; in Japan the current is upward on the Pacific side and downward on the Siberian side; in Austria it is upward in the north and downward in the south; in Great Britain upward in the east and downward in the west.” The results obtained for Great Britain differed considerably according as use was made of Rücker and Thorpe’s own district equations or of a series of general equations of the type subsequently utilized by Mathias. Tanakadate points out that the fact that his investigations give in each case a line of no current passing through the middle of the surveyed area, is calculated to throw doubt on the reality of the supposed earth-air currents, and he recommends a suspension of judgment.

§ 52. A question of interest, and bearing a relationship to the Gaussian analysis, is the law of variation of the magnetic elements with height above sea-level. If F represent the value at sea-level, and F + δF that at height h, of any component of force answering to Gaussian constants of the nth order, then 1 + δF/F = (1 + h/R)−n−2, where R is the earth’s radius. Thus at heights of only a few miles we have very approximately δF/F = −(n + 2) h/R. As we have seen, the constants of the first order are much the most important, thus we should expect as a first approximation δX/X = δY/Y = δZ/Z = −3h/R. This equation gives the same rate of decrease in all three components, and so no change in declination or inclination. Liznar^[82] compared this equation with the observed results of his Austrian survey, subdividing his stations into three groups according to altitude. He considered the agreement not satisfactory. It must be remembered that the Gaussian analysis, especially when only lower order terms are retained, applies only to the earth’s field freed from local disturbances. Now observations at individual high level stations may be seriously influenced not merely by regional disturbances common to low level stations, but by magnetic material in the mountain itself. A method of arriving at the vertical change in the elements, which theoretically seems less open to criticism, has been employed by A. Tanakadate.^[64] If we assume that a potential exists, or if admitting the possibility of earth-air currents we assume their effort negligible, we have dX/dz = dZ/dx, dY/dz = dZ/dy. Thus from the observed rates of change of the vertical component of force along the parallels of latitude and longitude, we can deduce the rate of change in the vertical direction of the two rectangular components of horizontal force, and thence the rates of change of the horizontal force and the declination. Also we have dZ/dz = 4πρ − (dX/dx + dY/dy), where ρ represents the density of free magnetism at the spot. The spot being above ground we may neglect ρ, and thus deduce the variation in the vertical direction of the vertical component from the observed variations of the two horizontal components in their own directions. Tanakadate makes a comparison of the vertical variations of the magnetic elements calculated in the two ways, not merely for Japan, but also for Austria-Hungary and Great Britain. In each country he took five representative points, those for Great Britain being the central stations of five of Rücker and Thorpe’s districts. Table XLVII. gives the mean of the five values obtained. By method (i.) is meant the formula involving 3h/R, by method (ii.) Tanakadate’s method as explained above. H, V, D, and I are used as defined in § 5. In the case of H and V unity represents 1γ.

Table XLVII.—Change per Kilometre of Height.

	Great Britain.		Austria-Hungary.		Japan.
Method.	(i.)	(ii.)	(i.)	(ii.)	(i.)	(ii.)
H	− 8.1	− 6.7	−10.1	− 8.7	−13.9	−14.0
V	−21.2	−19.4	−19.0	−18.1	−17.1	−17.4
D (west)	· ·	− 0′.04	· ·	+ 0′.10	· ·	− 0′.27
I	· ·	− 0′.05	· ·	− 0′.06	· ·	− 0′.01

The − sign in Table XLVII. denotes a decrease in the numerical values of H, V and I, and a diminution in westerly declination. If we except the case of the westerly component of force—not shown in the table—the accordance between the results from the two methods in the case of Japan is extraordinarily close, and there is no very marked tendency for the one method to give larger values than the other. In the case of Great Britain and Austria the differences between the two sets of calculated values though not large are systematic, the 3h/R formula invariably showing the larger reduction with altitude in both H and V. Tanakadate was so satisfied with the accordance of the two methods in Japan, that he employed his method to reduce all observed Japanese values to sea-level. At a few of the highest Japanese stations the correction thus introduced into the value of H was of some importance, but at the great majority of the stations the corrections were all insignificant.

§ 53. Schuster^[83] has calculated a potential analogous to the Gaussian potential, from which the regular diurnal changes of the magnetic elements all over the earth may be derived. From the mean summer and winter diurnal variations of the northerly and easterly components of force during Schuster’s Diurnal Variation Potential. 1870 at St Petersburg, Greenwich, Lisbon and Bombay, he found the values of 8 constants analogous to Gaussian constants; and from considerations as to the hours of occurrence of the maxima and minima of vertical force, he concluded that the potential, unlike the Gaussian, must proceed in positive powers of r, and so answer to forces external to the earth. Schuster found, however, that the calculated amplitudes of the diurnal vertical force inequality did not accord well with observation; and his conclusion was that while the original cause of the diurnal variation is external, and consists probably of electric currents in the atmosphere, there are induced currents inside the earth, which increase the horizontal components of the diurnal inequality while diminishing the vertical. The problem has also been dealt with by H. Fritsche,^[84] who concludes, in opposition to Schuster, that the forces are partly internal and partly external, the two sets being of fairly similar magnitude. Fritsche repeats the criticism (already made in the last edition of this encyclopaedia) that Schuster’s four stations were too few, and contrasts their number with the 27 from which his own data were derived. On the other hand, Schuster’s data referred to one and the same year, whereas Fritsche’s are from epochs varying from 1841 to 1896, and represent in some cases a single year’s observations, in other cases means from several years. It is clearly desirable that a fresh calculation should be made, using synchronous data from a considerable number of well distributed stations; and it should be done for at least two epochs, one representing large, the other small sun-spot frequency. The year 1870 selected by Schuster had, as it happened, a sun-spot frequency which has been exceeded only once since 1750; so that the magnetic data which he employed were far from representative of average conditions.

§ 54. It was discovered by Folgheraiter^[85] that old vases from Etruscan and other sources are magnetic, and from combined observation and experiment he concluded that they acquired their magnetization when cooling after being baked, and retained it unaltered. From experiments, he derived Magnetization of Vases, &c. formulae connecting the magnetization shown by new clay vases with their orientation when cooling in a magnetic field, and applying these formulae to the phenomena observed in the old vases he calculated the magnetic dip at the time and place of manufacture. His observations led him to infer that in Central Italy inclination was actually southerly for some centuries prior to 600 B.C., when it changed sign. In 400 B.C. it was about 20°N.; since 100 B.C. the change has been relatively small. L. Mercanton^[86] similarly investigated the magnetization of baked clay vases from the lake dwellings of Neuchatel, whose epoch is supposed to be from 600 to 800 B.C. The results he obtained were, however, closely similar to those observed in recent vases made where the inclination was about 63°N., and he concluded in direct opposition to Folgheraiter that inclination in southern Europe has not undergone any very large change during the last 2500 years. Folgheraiter’s methods have been extended to natural rocks. Thus B. Brunhes^[87] found several cases of clay metamorphosed by adjacent lava flows and transformed into a species of natural brick. In these cases the clay has a determinate direction of magnetization agreeing with that of the volcanic rock, so it is natural to assume that this direction coincided with that of the dip when the lava flow occurred. In drawing inferences, allowance must of course be made for any tilting of the strata since the volcanic outburst. From one case in France in the district of St Flour, where the volcanic action is assigned to the Miocene Age, Brunhes inferred a southerly dip of some 75°. Until a variety of cases have been critically dealt with, a suspension of judgment is advisable, but if the method should establish its claims to reliability it obviously may prove of importance to geology as well as to terrestrial magnetism.

§ 55. Magnetic phenomena in the polar regions have received considerable attention of late years, and the observed results are of so exceptional a character as to merit separate consideration. One feature, the large amplitude of the regular diurnal inequality, is already illustrated by the data for Jan Polar Phenomena. Mayen and South Victoria Land in Tables VIII. to XI. In the case, however, of declination allowance must be made for the small size of H. If a force F perpendicular to the magnetic meridian causes a change ΔD in D then ΔD = F/H. Thus at the “Discovery’s” winter quarters in South Victoria Land, where the value of H is only about 0.36 of that at Kew, a change of 45′ in D would be produced by a force which at Kew would produce a change of only 16′. Another feature, which, however, may not be equally general, is illustrated by the data for Fort Rae and South Victoria Land in Table XVII. It will be noticed that it is the 24-hour term in the Fourier analysis of the regular diurnal inequality which is specially enhanced. The station in South Victoria Land—the winter quarters of the “Discovery” in 1902–1904—was at 77° 51′ S. lat.; thus the sun did not set from November to February (midsummer), nor rise from May to July (midwinter). It might not thus have been surprising if there had been an outstandingly large seasonal variation in the type of the diurnal inequality. As a matter of fact, however, the type of the inequality showed exceptionally small variation with the season, and the amplitude remained large throughout the whole year. Thus, forming diurnal inequalities for the three midsummer months and for the three midwinter months, we obtain the following amplitudes for the range of the several elements^[88]:—

	D.	H.	V.	I.
Midsummer	64′.1	57γ	58γ	2′.87
Midwinter	26′.8	25γ	18γ	1′.23

The most outstanding phenomenon in high latitudes is the frequency and large size of the disturbances. At Kew, as we saw in § 25, the absolute range in D exceeds 20′ on only 12% of the total number of days. But at the “Discovery’s” winter quarters, about sun-spot minimum, the range exceeded 1° on 70%, 2° on 37%, and 3° on fully 15% of the total number of days. One day in 25 had a range exceeding 4°. During the three midsummer months, only one day out of 111 had a range under 1°, and even at midwinter only one day in eight had a range as small as 30′. The H range at the “Discovery’s” station exceeded 100γ on 40% of the days, and the V range exceeded 100γ on 32% of the days.

The special tendency to disturbance seen in equinoctial months in temperate latitudes did not appear in the “Discovery’s” records in the Antarctic. D ranges exceeding 3° occurred on 11% of equinoctial days, but on 40% of midsummer days. The preponderance of large movements at midsummer was equally apparent in the other elements. Thus the percentage of days having a V range over 200γ was 21 at midsummer, as against 3 in the four equinoctial months.

At the “Discovery’s” station small oscillations of a few minutes’ duration were hardly ever absent, but the character of the larger disturbances showed a marked variation throughout the 24 hours. Those of a very rapid oscillatory character were especially numerous in the morning between 4 and 9 a.m. In the late afternoon and evening disturbances of a more regular type became prominent, especially in the winter months. In particular there were numerous occurrences of a remarkably regular type of disturbance, half the total number of cases taking place between 7 and 9 p.m. This “special type of disturbance” was divisible into two phases, each lasting on the average about 20 minutes. During the first phase all the elements diminished in value, during the second phase they increased. In the case of D and H the rise and fall were about equal, but the rise in V was about 31/2 times the preceding fall. The disturbing force—on the north pole—to which the first phase might be attributed was inclined on the average about 5°1/2 below the horizon, the horizontal projection of its line of action being inclined about 41°1/2 to the north of east. The amplitude and duration of the disturbances of the “special type” varied a good deal; in several cases the disturbing force considerably exceeded 200γ. A somewhat similar type of disturbance was observed by Kr. Birkeland^[89] at Arctic stations also in 1902–1903, and was called by him the “polar elementary” storm. Birkeland’s record of disturbances extends only from October 1902 to March 1903, so it is uncertain whether “polar elementary” storms occur during the Arctic summer. Their usual time of occurrence seems to be the evening. During their occurrence Birkeland found that there was often a great difference in amplitude and character between the disturbances observed at places so comparatively near together as Iceland, Nova Zembla and Spitzbergen. This led him to assign the cause to electric currents in the Arctic, at heights not exceeding a few hundred kilometres, and he inferred from the way in which the phenomena developed that the seat of the disturbances often moved westward, as if related in some way to the sun’s position. Contemporaneously with the “elementary polar” storms in the Arctic Birkeland found smaller but distinct movements at stations all over Europe; these could generally be traced as far as Bombay and Batavia, and sometimes as far as Christchurch, New Zealand. Chree,^[88] on the other hand, working up the 1902–1904 Antarctic records, discovered that during the larger disturbances of the “special type” corresponding but much smaller movements were visible at Christchurch, Mauritius, Kolaba, and even at Kew. He also found that in the great majority of cases the Antarctic curves were specially disturbed during the times of Birkeland’s “elementary polar” storms, the disturbances in the Arctic and Antarctic being of the same order of magnitude, though apparently of considerably different type.

Examining the more prominent of the sudden commencements of magnetic disturbances in 1902–1903 visible simultaneously in the curves from Kew, Kolaba, Mauritius and Christchurch, Chree found that these were all represented in the Antarctic curves by movements of a considerably larger size and of an oscillatory character. In a number of cases Birkeland observed small simultaneous movements in the curves of his co-operating stations, which appeared to be at least sometimes decidedly larger in the equatorial than the northern temperate stations. These he described as “equatorial” perturbations, ascribing them to electric currents in or near the plane of the earth’s magnetic equator, at heights of the order of the earth’s radius. It was found, however, by Chree that in many, if not all, of these cases there were synchronous movements in the Antarctic, similar in type to those which occurred simultaneously with the sudden commencements of magnetic storms, and that these Antarctic movements were considerably larger than those described by Birkeland at the equatorial stations. This result tends of course to suggest a somewhat different explanation from Birkeland’s. But until our knowledge of facts has received considerable additions all explanations must be of a somewhat hypothetical character.

In 1831 Sir James Ross^[90] observed a dip of 89° 59′ at 70° 5′ N., 96° 46′ W., and this has been accepted as practically the position of the north magnetic pole at the time. The position of the south magnetic pole in 1840 as deduced from the Antarctic observations made by the “Erebus” and Magnetic Poles. “Terror” expedition is shown in Sabine’s chart as about 73° 30′ S., 147° 30′ E. In the more recent chart in J. C. Adams’s Collected Papers, vol. 2, the position is shown as about 73° 40′ S., 147° 7′ E. Of late years positions have been obtained for the south magnetic pole by the “Southern Cross” expedition of 1898–1900 (A), by the “Discovery” in 1902–1904 (B), and by Sir E. Shackleton’s expedition 1908–1909 (C). These are as follow:

(A)	72° 40′ S.,	152° 30′ E.
(B)	72° 51′ S.,	156° 25′ E.
(C)	72° 25′ S.,	155° 16′ E.

Unless the diurnal inequality vanishes in its neighbourhood, a somewhat improbable contingency considering the large range at the “Discovery’s” winter quarters, the position of the south magnetic pole has probably a diurnal oscillation, with an average amplitude of several miles, and there is not unlikely a larger annual oscillation. Thus even apart from secular change, no single spot of the earth’s surface can probably claim to be a magnetic pole in the sense popularly ascribed to the term. If the diurnal motion were absolutely regular, and carried the point where the needle is vertical round a closed curve, the centroid of that curve—though a spot where the needle is never absolutely vertical—would seem to have the best claim to the title. It should also be remembered that when the dip is nearly 90° there are special observational difficulties. There are thus various reasons for allowing a considerable uncertainty in positions assigned to the magnetic poles. Conclusions as to change of position of the south magnetic pole during the last ten years based on the more recent results (A), (B) and (C) would, for instance, possess a very doubtful value. The difference, however, between these recent positions and that deduced from the observations of 1840–1841 is more substantial, and there is at least a moderate probability that a considerable movement towards the north-east has taken place during the last seventy years.

See publications of individual magnetic observatories, more especially the Russian (Annales de l’Observatoire Physique Central), the French (Annales du Bureau Central Météorologique de France), and those of Kew, Greenwich, Falmouth, Stonyhurst, Potsdam, Wilhelmshaven, de Bilt, Uccle, O’Gyalla, Prague, Pola, Coimbra, San Fernando, Capo di Monte, Tiflis, Kolaba, Zi-ka-wei, Hong-Kong, Manila, Batavia, Mauritius, Agincourt (Toronto), the observatories of the U.S. Coast and Geodetic Survey, Rio de Janeiro, Melbourne.

In the references below the following abbreviations are used: B.A. = British Association Reports; Batavia = Observations made at the Royal . . . Observatory at Batavia; M.Z. = Meteorologische Zeitschrift, edited by J. Hann and G. Hellman; P.R.S. = Proceedings of the Royal Society of London; P.T. = Philosophical Transactions; R. = Repertorium für Meteorologie, St Petersburg; T.M. = Terrestrial Magnetism, edited by L. A. Bauer; R.A.S. Notices = Monthly Notices of the Royal Astronomical Society. Treatises are referred to by the numbers attached to them; e.g. (1) p. 100 means p. 100 of Walker’s Terrestrial Magnetism.

¹ E. Walker, Terrestrial and Cosmical Magnetism (Cambridge and London, 1866). ^1a H. Lloyd, A Treatise on Magnetism General and Terrestrial (London, 1874). ² E. Mascart, Traité de magnétisme terrestre (Paris, 1900). ³ L. A. Bauer, United States Magnetic Declination Tables and Isogonic Charts, and Principal Facts relating to the Earth’s Magnetism (Washington, 1902). ⁴ Balfour Stewart, “Terrestrial Magnetism” (under “Meteorology”), Ency. brit. 9th ed. ⁵ C. Chree, “Magnetism, Terrestrial,” Ency. brit. 10th ed. ⁶M.Z. 1906, 23, p. 145. ⁷(3) p. 62. ⁸K. Akad. van Wetenschappen (Amsterdam, 1895; Batavia, 1899, &c.). ⁹ Atlas des Erdmagnetismus (Riga, 1903). ¹⁰ (1) p. 16, &c. ¹¹ Kolaba (Colaba) Magnetical and Meteorological Observations, 1896, Appendix Table II. ¹² (1) p. 21. ¹³ Report for 1906, App. 4, see also (3) p. 102. ¹⁴(1) p. 166. ¹⁵ Ergebnisse der mag. Beobachtungen in Potsdam, 1901, p. xxxvi. ¹⁶ U.S. Coast and Geodetic Survey Report for 1895, App. I, &c. ¹⁷ T.M. 1, pp. 62, 89, and 2, p. 68. ¹⁸ (3) p. 45. ¹⁹ Die Elemente des Erdmagnetismus, pp. 104-108. ²⁰ Zur täglichen Variation der mag. Deklination (aus Hefl II. des Archivs des Erdmagnetismus) (Potsdam, 1906). ²¹ M.Z. 1888, 5, p. 225. ²²M.Z. 1904, 21, p. 129. ²³P.T. 202 A, p. 335. ^23aComb. Phil. Soc. Trans. 20, p. 165. ²⁴P.T. 208 A, p. 205. ²⁵P.T. 203 A, p. 151. ²⁶P.T. 171. p. 541; P.R.S. 63, p. 64. ²⁷R.A.S. Notices 60, p. 142. ²⁸Rendiconti del R. Ist. Lomb. 1902, Series II. vol. 35. ²⁹R. 1889, vol. 12, no. 8. ³⁰B.A. Report, 1898, p. 80. ³¹P.R.S. (A) 79, p. 151. ³²P.T. 204 A, p. 373. ³³Ann. du Bureau Central Météorologique, année 1897, 1 Mem. p. B65. ³⁴P.T. 161, p. 307. ³⁵M.Z. 1895, 12, p. 321. ^35aP.T. 1851, p. 123; and 1852, p. 103, see also (4) § 38. ³⁶P.T. 159, p. 363. ³⁷(1) p. 92. ³⁸R.A.S. Notices 65, p. 666. ³⁹R.A.S. Notices, 65, pp. 2 and 538. ⁴⁰K. Akad. van Wetenschappen (Amsterdam, 1906) p. 266. ⁴¹R.A.S. Notices 65, p. 520. ⁴²B.A. Reports, 1880, p. 201 and 1881, p. 463. ⁴³Anhang Ergebnisse der mag. Beob. in Potsdam, 1896. ⁴⁴M.Z. 1899, 16, p. 385. ⁴⁵P.T. 166, p. 387. ⁴⁶Trans. Can. Inst. 1898–1899, p. 345, and Proc. Roy. Ast. Soc. of Canada, 1902–1903, p. 74, 1904, p. xiv., &c. ⁴⁷R.A.S. Notices 65, p. 186. ⁴⁸T.M. 10, p. 1. ⁴⁹Expédition norvégienne de 1899–1900 (Christiania, 1901). ⁵⁰Thèses présentées à la Faculté des Sciences (Paris, 1903). ⁵¹Nat. Tijdschrift voor Nederlandsch-Indië, 1902, p. 71. ⁵²Wied. Ann. 1882, p. 336. ⁵³Sitz. der k. preuss. Akad. der Wiss., 24th June 1897, &c. ⁵⁴T.M. 12, p. 1. ⁵⁵P.T. 143, p. 549; St Helena Observations, vol. ii., p. cxlvi., &c., (1) § 62. ⁵⁶Trans. R.S.E. 24, p. 669. ⁵⁷P.T. 178 A, p. 1. ⁵⁸Batavia, vol. 16, &c. ⁵⁹Batavia, Appendix to vol. 26. ⁶⁰R. vol. 17, no. 1. ⁶¹T.M. 3, p. 1, &c. ⁶²P.T. 181 A, p. 53 and 188 A. ⁶³Ann. du Bureau Central Mét. vol. i. for years 1884 and 1887 to 1895. ⁶⁴Ann. dell’ Uff. Centrale Met. e Geod. vol. 14, pt. i. p. 57. ⁶⁵A Magnetic Survey of the Netherlands for the Epoch 1st Jan. 1891 (Rotterdam, 1895). ⁶⁶Kg. Svenska Vet. Akad. Handlingar, 1895, vol. 27, no. 7. ⁶⁷Denkschriften der math. naturwiss. Classe der k. Akad. des Wiss. (Wien), vols. 62 and 67. ⁶⁸Journal of the College of Science, Tōkyō, 1904, vol. 14. ⁶⁹Ann. de l’observatoire . . . de Toulouse, 1907, vol. 7. ⁷⁰Ann. du Bureau Central Mét. 1897, I. p. B36. ⁷¹T.M. 7, p. 74. ⁷²Bull. Imp. Univ. Odessa 85, p. 1, and T.M. 7, p. 67. ⁷³P.T. 187 A, p. 345. ⁷⁴P.R.S. 76 A, p. 181. ⁷⁵Bull. Soc. Imp. des Naturalistes de Moskau, 1893, no. 4, p. 381, and T.M. 1, p. 50. ⁷⁶Forsch. zur deut. Landes- u. Volkskunde, 1898, Bd. xi, 1, and T.M. 3, p. 77. ⁷⁷P.R.S. 76 A, p. 507. ⁷⁸Adams, Scientific Papers, II. p. 446. ⁷⁹B.A. Report for 1898, p. 109. ⁸⁰Abhand. der bayer, Akad. der Wiss., 1895, vol. 19. ⁸¹Sitz. k. Akad. der Wiss. (Berlin), 1897, no. xviii., also T.M. 3, p. 191. ⁸²T.M. 2, p. 11. ⁸³Die Elemente des Erdmagnetismus (St Petersburg, 1899), p. 103. ⁸⁴ T.M. 9, p. 113. ⁸⁵ T.M. 1, p. 77, and Nature, 57, pp. 160 and 180. ⁸⁶ M.Z. 15, p. 175. ^86a Sitz. der k. k. Akad. der Wiss. Wien, math. nat. Classe, 1898, Bd. cvii., Abth. ii. ⁸⁷ P.T. (A) 180, p. 467. ⁸⁸ Die Tägliche Periode der erdmagnetischen Elemente (St Petersburg, 1902). ⁸⁹ R. Accad. Lincei Atti, viii. 1899, pp. 69, 121, 176, 269 and previous volumes, see also Séances de la Soc. Franc. de Physique, 1899, p. 118. ⁹⁰ Bull. Soc. Vaud., Sc. Nat. 1906, 42, p. 225. ⁹¹ Comptes rendus, 1905, 141, p. 567. ⁹² National Antarctic Expedition 1901–1904, “Magnetic Observations.” ⁹³ The Norwegian Aurora Polaris Expedition 1902–1903, vol. i. ⁹⁴ (1) p. 163.  (C. Ch.) 

---

1. E. Walker, Terrestrial and Cosmical Magnetism (Cambridge and London, 1856).
2. L. A. Bauer, United States Magnetic Declination Tables and Isogonic Charts, and Principal Facts relating to the Earth’s Magnetism (Washington, 1902).
3. (3) p. 62.
4. K. Akad. van Wetenschappen (Amsterdam, 1895; Batavia, 1899, &c.).
5. Atlas des Erdmagnetismus (Riga, 1903).
6. (1) p. 16, &c.
7. Kolaba (Colaba) Magnetical and Meteorological Observations, 1896. Appendix Table II.
8. (1) p. 21.
9. Report for 1906, App. 4, see also (3) p. 102.
10. (1) p. 166.
11. Ergebnisse der mag. Beobachtungen in Potsdam, 1901, p. xxxvi.
12. U.S. Coast and Geodetic Survey Report for 1895, App. 1, &c.
13. T.M. 1, pp. 62, 89, and 2, p. 68.
14. (3) p. 45.
15. Die Elemente des Erdmagnetismus, pp. 104.108.
16. Zur täglichen Variation der mag. Deklination (aus Heft II. des Archivs des Erdmagnetismus) (Potsdam, 1906).
17. M.Z. 1888, 5, p. 225.
18. M.Z. 1904, 21, p. 129.
19. Comb. Phil. Soc. Trans. 20, p. 165. P.T. 202 A, p. 335.
20. P.T. 208 A, p. 205. P.T. 208 A, p. 205. P.T. 208 A, p. 205.
21. P.T. 203 A, p. 151. P.T. 203 A, p. 151.
22. P.T. 171. p. 541; P.R.S. 63, p. 64.
23. R.A.S. Notices 60, p. 142. R.A.S. Notices 60, p. 142.
24. Rendiconti del R. Ist. Lomb. 1902, Series II. vol. 35.
25. R. 1889, vol. 12, no. 8.
26. B.A. Report, 1898, p. 80. B.A. Report, 1898, p. 80.
27. P.R.S. (A) 79, p. 151.
28. P.T. 204 A, p. 373.
29. Ann. du Bureau Central Météorologique, année 1897, 1 Mem. p. B65. Ann. du Bureau Central Météorologique, année 1897, 1 Mem. p. B65.
30. P.T. 161, p. 307.
31. M.Z. 1895, 12, p. 321. M.Z. 1895, 12, p. 321.
32. P.T. 159, p. 363.
33. p. 92.
34. R.A.S. Notices 65, p. 666.
35. R.A.S. Notices, 65, pp. 2 and 538. R.A.S. Notices, 65, pp. 2 and 538.
36. K. Akad. van Wetenschappen (Amsterdam, 1906) p. 266.
37. R.A.S. Notices 65, p. 520.
38. B.A. Reports, 1880, p. 201 and 1881, p. 463.
39. Anhang Ergebnisse der mag. Beob. in Potsdam, 1896.
40. M.Z. 1899, 16, p. 385.
41. P.T. 166, p. 387.
42. Trans. Can. Inst. 1898–1899, p. 345, and Proc. Roy. Ast. Soc. of Canada, 1902–1903, p. 74, 1904, p. xiv., &c.
43. R.A.S. Notices 65, p. 186.
44. T.M. 10, p. 1.
45. Expédition norvégienne de 1899–1900 (Christiania, 1901). Expédition norvégienne de 1899–1900 (Christiania, 1901).
46. Thèses présentées à la Faculté des Sciences (Paris, 1903).
47. Nat. Tijdschrift voor Nederlandsch-Indië, 1902, p. 71.
48. Wied. Ann. 1882, p. 336.
49. Sitz. der k. preuss. Akad. der Wiss., 24th June 1897, &c.
50. T.M. 12, p. 1.
51. P.T. 143, p. 549; St Helena Observations, vol. ii.,
52. p. cxlvi., &c., (1) § 62. Trans. R.S.E. 24, p. 669.
53. P.T. 178 A, p. 1.
54. Batavia, vol. 16, &c.
55. Batavia, Appendix to vol. 26.
56. R. vol. 17, no. 1.
57. T.M. 3, p. 1, &c.
58. P.T. 181 A, p. 53 and 188 A.
59. Ann. du Bureau Central Mét. vol. i. for years 1884 and 1887 to 1895.
60. Ann. dell’ Uff. Centrale Met. e Geod. vol. 14, pt. i. p. 57.
61. A Magnetic Survey of the Netherlands for the Epoch 1st Jan. 1891 (Rotterdam, 1895).
62. Kg. Svenska Vet. Akad. Handlingar, 1895, vol. 27, no. 7.
63. Denkschriften der math. naturwiss. Classe der k. Akad. des Wiss. (Wien), vols. 62 and 67.
64. Journal of the College of Science, Tōkyō, 1904, vol. 14. Journal of the College of Science, Tōkyō, 1904, vol. 14. Journal of the College of Science, Tōkyō, 1904, vol. 14.
65. Ann. de l’observatoire . . . de Toulouse, 1907, vol. 7.
66. Ann. du Bureau Central Mét. 1897, I. p. B36.
67. T.M. 7, p. 74.
68. Bull. Imp. Univ. Odessa 85, p. 1, and T.M. 7, p. 67.
69. P.T. 187 A, p. 345.
70. P.R.S. 76 A, p. 181.
71. Bull. Soc. Imp. des Naturalistes de Moskau, 1893, no. 4, p. 381, and T.M. 1, p. 50.
72. Forsch. zur deut. Landes- u. Volkskunde, 1898, Bd. xi, 1, and T.M. 3, p. 77.
73. P.R.S. 76 A, p. 507.
74. Adams, Scientific Papers, II. p. 446.
75. B.A. Report for 1898, p. 109.
76. Abhand. der bayer, Akad. der Wiss., 1895, vol. 19.
77. Sitz. k. Akad. der Wiss. (Berlin), 1897, no. xviii., also T.M. 3, p. 191.
78. T.M. 2, p. 11.
79. Die Elemente des Erdmagnetismus (St Petersburg, 1899), p. 103.
80. T.M. 9, p. 113.
81. T.M. 1, p. 77, and Nature, 57, pp. 160 and 180.
82. M.Z. 15, p. 175.
83. P.T. (A) 180, p. 467.
84. Die Tägliche Periode der erdmagnetischen Elemente (St Petersburg, 1902).
85. R. Accad. Lincei Atti, viii. 1899, pp. 69, 121, 176, 269 and previous volumes, see also Séances de la Soc. Franc. de Physique, 1899, p. 118.
86. Bull. Soc. Vaud., Sc. Nat. 1906, 42, p. 225.
87. Comptes rendus, 1905, 141, p. 567.
88. National Antarctic Expedition 1901–1904, “Magnetic Observations.” National Antarctic Expedition 1901–1904, “Magnetic Observations.”
89. The Norwegian Aurora Polaris Expedition 1902–1903, vol. i.
90. (1) p. 163.

1. For explanation of these numbers, see end of article.