§ 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[80] 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[81] and Bauer[82] 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[83] treating the problem similarly, but for two epochs, 1842 and 1885, got conspicuously different results for the two epochs, Bauer[84] 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,[85] and to Austria by Liznar,[86] but with negative results. The question has also been considered in detail by Tanakadate[68] 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) dx dy.
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[86a] 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.[68] 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.