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[1] and others with considerable success. The
following examples, due to G. W. Littlehales,[2] 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[3] 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[4] 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,
