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
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[1] 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[2] and Bauer[3] 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[4] treating the problem similarly, but for two epochs, 1842 and 1885, got conspicuously different results for the two epochs, Bauer[5] 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 130 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,[6] and to Austria by Liznar,[7] but with negative results. The question has also been considered in detail by Tanakadate[8] 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.
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[7] compared this equation with the observed results of his Austrian survey, subdividing his stations into three groups according
- ↑ Abhand. der bayer, Akad. der Wiss., 1895, vol. 19.
- ↑ Sitz. k. Akad. der Wiss. (Berlin), 1897, no. xviii., also T.M. 3, p. 191.
- ↑ T.M. 2, p. 11.
- ↑ Die Elemente des Erdmagnetismus (St Petersburg, 1899), p. 103.
- ↑ T.M. 9, p. 113.
- ↑ T.M. 1, p. 77, and Nature, 57, pp. 160 and 180.
- ↑ 7.0 7.1 M.Z. 15, p. 175.
- ↑ Journal of the College of Science, Tōkyō, 1904, vol. 14.
