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[69] 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 − .000239l²,
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′½ for inclination, and 70γ for horizontal force. The applications by Mathias[69] 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 Local Disturbances. peculiar to the neighbourhood. Subtracting the forces derived from the formulae from those observed, we 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[70] and by E. Leyst.[71] 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.[72] 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°½ 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 1¼ 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.[73] 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[74]—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.[75] Again, Eschenhagen[76] 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[72] 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[77] 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 Gaussian Potential and Constants. 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 μ = cos (½π − l), where l is latitude; and also let
| Hmn = (1 − µ2)1/2m [ µn−m − | (n − m) (n − m − 1) | µn−m−2 + ... ], |
| 2 (2n − 1) |