Page:EB1911 - Volume 17.djvu/396

(48° N. lat. 33° E. long.) was elaborately surveyed by Paul Passalsky.^[1] 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.^[2] 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^[3]—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.^[4] Again, Eschenhagen^[5] 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^[1] 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^[6] 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^[7] 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.^[8]

§ 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

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1. Bull. Imp. Univ. Odessa 85, p. 1, and T.M. 7, p. 67.
2. P.T. 187 A, p. 345.
3. P.R.S. 76 A, p. 181.
4. Bull. Soc. Imp. des Naturalistes de Moskau, 1893, no. 4, p. 381, and T.M. 1, p. 50.
5. Forsch. zur deut. Landes- u. Volkskunde, 1898, Bd. xi, 1, and T.M. 3, p. 77.
6. P.R.S. 76 A, p. 507.
7. Adams, Scientific Papers, II. p. 446.
8. B.A. Report for 1898, p. 109.