The question whether lunar influence increases with sun-spot frequency is obviously of considerable theoretical interest. Balfour Stewart in the 9th edition of this encyclopaedia gave some data indicating an appreciably enhanced lunar influence at Trivandrum during years of sun-spot maximum, but he hesitated to accept the result as finally proved. Figee recently investigated this point at Batavia, but with inconclusive results. Attempts have also been made to ascertain how lunar influence depends on the moon’s declination and phase, and on her distance from the earth. The difficulty in these investigations is that we are dealing with a small effect, and a very long series of data would be required satisfactorily to eliminate other periodic influences.§ 41. From an analysis of seventeen years data at St Petersburg and Pavlovsk, Leyst60concluded that all the principal planets sensibly influence the earth’s magnetism. According to his figures, all the planets except Mercury—whose influencePlanetary Influence.he found opposite to that of the others—when nearest the earth tended to deflect the declination magnet at St Petersburg to the west, and also increased the range of the diurnal inequality of declination, the latter effect being the more conspicuous. Schuster,61who has considered the evidence advanced by Leyst from the mathematical standpoint, considers it to be inconclusive.§ 42. The best way of carrying out a magnetic survey depends on where it has to be made and on the object in view. The object that probably still comes first in importance is a knowledge of the declination, of sufficient accuracy for navigationMagnetic Surveys.in all navigable waters. One might thus infer that magnetic surveys consist mainly of observations at sea. This cannot however be said to be true of the past, whatever it may be of the future, and this for several reasons. Observations at sea entail the use of a ship, specially constructed so as to be free from disturbing influence, and so are inherently costly; they are also apt to be of inferior accuracy. It might be possible in quiet weather, in a large vessel free from vibration, to observe with instruments of the highest precision such as a unifilar magnetometer, but in the ordinary surveying ship apparatus of less sensitiveness has to be employed. The declination is usually determined with some form of compass. The other elements most usually found directly at sea are the inclination and the total force, the instrument employed being a special form of inclinometer, such as the Fox circle, which was largely used by Ross in the Antarctic, or in recent years the Lloyd-Creak. This latter instrument differs from the ordinary dip-circle fitted for total force observations after H. Lloyd’s method mainly in that the needles rest in pivots instead of on agate edges. To overcome friction a projecting pin on the framework is scratched with a roughened ivory plate.The most notable recent example of observations at sea is afforded by the cruises of the surveying ships “Galilee” and “Carnegie” under the auspices of the Carnegie Institution of Washington, which includes in its magnetic programme a general survey. To see where the ordinary land survey assists navigation, let us take the case of a country with a long seaboard. If observations were taken every few miles along the coast results might be obtained adequate for the ordinary wants of coasting steamers, but it would be difficult to infer what the declination would be 50 or even 20 miles off shore at any particular place. If, however, the land area itself is carefully surveyed, one knows the trend of the lines of equal declination, and can usually extend them with considerable accuracy many miles out to sea. One also can tell what places if any on the coast suffer from local disturbances, and thus decide on the necessity of special observations. This is by no means the only immediately useful purpose which is or may be served by magnetic surveys on land. In Scandinavia use has been made of magnetic observations in prospecting for iron ore. There are also various geological and geodetic problems to whose solution magnetic surveys may afford valuable guidance. Among the most important recent surveys may be mentioned those of the British Isles by A. Rücker and T. E. Thorpe,62of France and Algeria by Moureaux,63of Italy by Chistoni and Palazzo,64of the Netherlands by Van Ryckevorsel,65of South Sweden by Carlheim Gyllenskiöld,66of Austria-Hungary by Liznar,67of Japan by Tanakadate,68of the East Indies by Van Bemmelen, and South Africa by J. C. Beattie. A survey of the United States has been proceeding for a good many years, and many results have appeared in the publications of the U.S. Coast and Geodetic Survey, especially Bauer’sMagnetic Tables and Magnetic Charts, 1908. Additions to our knowledge may also be expected from surveys of India, Egypt and New Zealand.For the satisfactory execution of a land survey, the observers must have absolute instruments such as the unifilar magnetometer and dip circle, suitable for the accurate determination of the magnetic elements, and they must be able to fix the exact positions of the spots where observations are taken. If, as usual, the survey occupies several years, what is wanted is the value of the elements not at the actual time of observation, but at some fixed epoch, possibly some years earlier or later. At a magnetic observatory, with standardized records, the difference between the values of a magnetic element at any two specified instants can be derived from the magnetic curves. But at an ordinary survey station, at a distance from an observatory, the information is not immediately available. Ordinarily the reduction to a fixed epoch is done in at least two stages, a correction being applied for secular change, and a second for the departure from the mean value for the day due to the regular diurnal inequality and to disturbance.The reduction to a fixed epoch is at once more easy and more accurate if the area surveyed contains, or has close to its borders, a well distributed series of magnetic observatories, whose records are comparable and trustworthy. Throughout an area of the size of France or Germany, the secular change between any two specified dates can ordinarily be expressed with sufficient accuracy by a formula of the typeδ = δ0+ a (l − l0) + b (λ − λ0) . . (i),where δ denotes secular change, l latitude and λ longitude, the letters with suffix0relating to some convenient central position. The constants δ0,a,bare to be determined from the observed secular changes at the fixed observatories whose geographical co-ordinates are accurately known. Unfortunately, as a rule, fixed observatories are few in number and not well distributed for survey purposes; thus the secular change over part at least of the area has usually to be found by repeating the observations after some years at several of the field stations. The success attending this depends on theexactitude with which the sites can be recovered, on the accuracy of the observations, and on the success with which allowance is made for diurnal changes, regular and irregular. It is thus desirable that the observations at repeat stations should be taken at hours when the regular diurnal changes are slow, and that they should not be accepted unless taken on days that prove to be magnetically quiet. Unless the secular change is exceptionally rapid, it will usually be most convenient in practice to calculate it from or to the middle of the month, and then to allow for the difference between the mean value for the month and the value at the actual hour of observation. There is here a difficulty, inasmuch as the latter part of the correction depends on the diurnal inequality, and so on the local time of the station. No altogether satisfactory method of surmounting this difficulty has yet been proposed. Rücker and Thorpe in their British survey assumed that the divergence from the mean value at any hour at any station might be regarded as made up of a regular diurnal inequality, identical with that at Kew when both were referred tolocaltime, and of a disturbance element identical with that existing at the same absolute time at Kew. Suppose, for instance, that at hourhG.M.T. the departure at Kew from the mean value for the month isd, then the corresponding departure from the mean at a station λ degrees west of Kew isd − e, whereeis the increase in the element at Kew due to the regular diurnal inequality between hourh− λ/15 and hourh. This procedure is simple, but is exposed to various criticisms. If we define a diurnal inequality as the result obtained by combining hourly readings from all the days of a month, we can assign a definite meaning to the diurnal inequality for a particular month of a particular year, and after the curves have been measured we can give exact numerical figures answering to this definition. But the diurnal inequality thus obtained differs, as has been pointed out, from that derived from a limited number of the quietest days of the month, not merely in amplitude but in phase, and the view that the diurnal changes on any individual day can be regarded as made up of a regular diurnal inequality of definite character and of a disturbance element is an hypothesis which is likely at times to be considerably wide of the mark. The extent of the error involved in assuming the regular diurnal inequality the same in the north of Scotland, or the west of Ireland, as in the south-east of England remains to be ascertained. As to the disturbance element, even if the disturbing force were of given magnitude and direction all over the British Isles—which we now know is often very far from the case—its effects would necessarily vary very sensibly owing to the considerable variation in the direction and intensity of the local undisturbed force. If observations were confined to hours at which the regular diurnal changes are slow, and only those taken on days of little or no disturbance were utilized, corrections combining the effects of regular and irregular diurnal changes could be derived from the records of fixed observations, supposed suitably situated, combined in formulae of the same type as (i).§ 43. The field results having been reduced to a fixed epoch, it remains to combine them in ways likely to be useful. In most cases the results are embodied in charts, usually of at least two kinds, one set showing only general features, the other the chief local peculiarities. Charts of the first kind resemble the world charts (figs. 1 to 4) in being free from sharp twistings and convolutions. In these the declination for instance at a fixed geographical position on a particular isogonal is to be regarded as really a mean from a considerable surrounding area.Various ways have been utilized for arriving at theseterrestrial isomagnetics—as Rücker and Thorpe call them—of which an elaborate discussion has been made by E. Mathias.69From a theoretical standpoint the simplest method is perhaps that employed by Liznar for Austria-Hungary. Letland λ represent latitude and longitude relative to a certain central station in the area. Then assume that throughout the area the value E of any particular magnetic element is given by a formulaE = E0+ al + bλ + cl² + dλ² + elλ,where E0, a, b, c, d, e are absolute constants to be determined from the observations. When determining the constants, we write for E in the equation the observed value of the element (corrected for secular change, &c.) at each station, and for l and λ the latitude and longitude of the station relative to the central station. Thus each station contributes an equation to assist in determining the six constants. They can thus be found by least squares or some simpler method. In Liznar’s case there were 195 stations, so that the labour of applying least squares would be considerable. This is one objection to the method. A second is that it may allow undesirably large weight to a few highly disturbed stations. In the case of the British Isles, Rücker and Thorpe employed a different method. The area was split up intodistricts. For each district a mean was formed of the observed values of each element, and the mean was assigned to an imaginary central station, whose geographical co-ordinates represented the mean of the geographical co-ordinates of the actual stations. Want of uniformity in the distribution of the stations may be allowed for by weighting the results. Supposing E0the value of the element found for the central station of a district, it was assumed that the value E at any actual station whose latitude and longitude exceeded those of the central station by l and λ was given by E = E0+ al + bλ, with a and b constants throughout the district. Having found E0, a and b, Rücker and Thorpe calculated values of the element for points defined by whole degrees of longitude (from Greenwich) and half degrees of latitude. Near the common border of two districts there would be two calculated values, of which the arithmetic mean was accepted.The next step was to determine by interpolation where isogonals—or other isomagnetic lines—cut successive lines of latitude. The curves formed by joining these successive points of intersection were calleddistrictlines or curves. Rücker and Thorpe’s next step was to obtain formulae by trial, giving smooth curves of continuous curvature—terrestrial isomagnetics—approximating as closely as possible to the district lines. The curves thus obtained had somewhat complicated formulae. For instance, the isogonals south of 54°.5 latitude were given for the epoch Jan. 1, 1891 byD = 18° 37′ + 18′.5 (l − 49.5) − 3′.5 cos {45° (l − 49.5) }+ {26′.3 + 1′.5 (l − 49.5) } (λ − 4) + 0′.01 (λ − 4)² (l − 54.5)²,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. Baillaud69have 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, wasD = 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 Mathias69of 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 disturbancesLocal 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 termedridge lines. The corresponding loci where the largest negative values occur were calledvalley 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 Moureaux70and by E. Leyst.71In 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.72The 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.73Within 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. Field74—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.75Again, Eschenhagen76detected 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 Passalsky72at 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. Claxton77found 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°1⁄3in 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 surfaceGaussian 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 letHmn= (1 − µ2)1/2m[µn−m−(n − m) (n − m − 1)µn−m−2+ ...],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 haveV / R = Σ (R / r)n+1[Hmn(gmncos mλ + hmnsin mλ) ]+ Σ (r / R)n[Hmn(gm−ncos mλ + hm−nsin mλ) ],where Σ denotes summation of m from 0 to n, followed by summation of n from 0 to ∞. In this equation gmn, &c. are constants, those with positive suffixes being what are generally termedGaussian 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 nthorder is a spherical harmonic of the nthdegree. If R be taken as unit length, as is not infrequent, the first order terms are given byV1= r−2[g10sin l + (g11cos λ + h11sin λ) cos l].The earth is in reality a spheroid, and in his elaborate work on the subject J. C. Adams78develops 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 surfaceX = −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 gmn, &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.1829Erman-Petersen.1830Gauss.1845Adams.1880Adams.1885Neumayer.1885Schmidt.1885Fritsche.g10+.32007+.32348+.32187+.31684+.31572+.31735+.31635g11+.02835+.03111+.02778+.02427+.02481+.02356+.02414h11−.06011−.06246−.05783−.06030−.06026−.05984−.05914Table 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.79§ 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:—g10g11h111842-1845+.32173+.02833−.058201880+.31611+.02470−.06071Some of the higher constants were relatively much more affected. Thus, on the hypotheses of 48 and of 24 constants respectively, the values obtained for g20in 1842-1845 were -.00127 and -.00057, and those obtained for h31in 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, gmnmust be replaced by gmn+ gm−n, and in any equation determining Z by gmn− {n/(n + 1)} gmn; similar remarks apply to hmnand 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 gmnand hmnfrom the X and Y or from the Z equations, when gm−nand hm−nare assumed zero. If the values so found differ, values can be found for gm−nand hm−nwhich will harmonize the two sets of equations. Adams gives the values obtained from the X, Y and the Z equations separately for theGaussian constants. The following are examples of the values thence deducible for the coefficients of the positive power series:—g−10g−11h−11g−40g−50g−601842-1845+.0018−.0002−.0014+.0064+.0072+.01241880−.0002−.0012+.0015−.0043−.0021−.0013Compared to g40, g50and g60the values here found for g−40, g−50and g−60are 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−11and 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 forConstants.NorthLatitude.WestLongitude.M/R3inG.C.S. units.° ′° ′1650H. Fritsche82 5042 55.32601836”78 2763 35.32621845J. C. Adams78 4464 20.32821880”78 2468 4.32341885Neumayer-Petersen and Bauer78 367 3.32241885Neumayer, Schmidt78 3468 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 byM = R3{ (g10)2+ (g11)2+ (h11)2}1/2,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 thenormalfield—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. Schmidt80Earth-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 Bezold81and Bauer82the 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. Fritsche83treating the problem similarly, but for two epochs, 1842 and 1885, got conspicuously different results for the two epochs, Bauer84has 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 anupwardlydirected 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+ 3615° to 30°−70−59−68+ 2− 62− 6330° to 45°+ 3+14−22+26− 11− 1445° to 60°−31−21+78+ 5+276+213In 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 of1⁄30ampere 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,85and to Austria by Liznar,86but with negative results. The question has also been considered in detail by Tanakadate68in discussing the magnetic survey of Japan. He makes the criticism that the taking of a line integral round theboundaryof 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 nthorder, 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. Liznar86acompared this equation with the observed results of his Austrian survey, subdividing his stations into three groups accordingto 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.68If 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.Great Britain.Austria-Hungary.Japan.Method.(i.)(ii.)(i.)(ii.)(i.)(ii.)H− 8.1− 6.7−10.1− 8.7−13.9−14.0V−21.2−19.4−19.0−18.1−17.1−17.4D (west)· ·− 0′.04· ·+ 0′.10· ·− 0′.27I· ·− 0′.05· ·− 0′.06· ·− 0′.01The − sign in Table XLVII. denotes a decrease in the numerical values of H, V and I, and a diminution in westerly declination. If we except the case of the westerly component of force—not shown in the table—the accordance between the results from the two methods in the case of Japan is extraordinarily close, and there is no very marked tendency for the one method to give larger values than the other. In the case of Great Britain and Austria the differences between the two sets of calculated values though not large are systematic, the 3h/R formula invariably showing the larger reduction with altitude in both H and V. Tanakadate was so satisfied with the accordance of the two methods in Japan, that he employed his method to reduce all observed Japanese values to sea-level. At a few of the highest Japanese stations the correction thus introduced into the value of H was of some importance, but at the great majority of the stations the corrections were all insignificant.§ 53. Schuster87has calculated a potential analogous to the Gaussian potential, from which the regular diurnal changes of the magnetic elements all over the earth may be derived. From the mean summer and winter diurnal variationsSchuster’s Diurnal Variation Potential.of the northerly and easterly components of force during 1870 at St Petersburg, Greenwich, Lisbon and Bombay, he found the values of 8 constants analogous to Gaussian constants; and from considerations as to the hours of occurrence of the maxima and minima of vertical force, he concluded that the potential, unlike the Gaussian, must proceed in positive powers ofr, and so answer to forces external to the earth. Schuster found, however, that the calculated amplitudes of the diurnal vertical force inequality did not accord well with observation; and his conclusion was that while the original cause of the diurnal variation is external, and consists probably of electric currents in the atmosphere, there are induced currents inside the earth, which increase the horizontal components of the diurnal inequality while diminishing the vertical. The problem has also been dealt with by H. Fritsche,88who concludes, in opposition to Schuster, that the forces are partly internal and partly external, the two sets being of fairly similar magnitude. Fritsche repeats the criticism (already made in the last edition of this encyclopaedia) that Schuster’s four stations were too few, and contrasts their number with the 27 from which his own data were derived. On the other hand, Schuster’s data referred to one and the same year, whereas Fritsche’s are from epochs varying from 1841 to 1896, and represent in some cases a single year’s observations, in other cases means from several years. It is clearly desirable that a fresh calculation should be made, using synchronous data from a considerable number of well distributed stations; and it should be done for at least two epochs, one representing large, the other small sun-spot frequency. The year 1870 selected by Schuster had, as it happened, a sun-spot frequency which has been exceeded only once since 1750; so that the magnetic data which he employed were far from representative of average conditions.§ 54. It was discovered by Folgheraiter89that old vases from Etruscan and other sources are magnetic, and from combined observation and experiment he concluded that they acquired their magnetization when cooling after being baked, andMagnetization of Vases, &c.retained it unaltered. From experiments, he derived formulae connecting the magnetization shown by new clay vases with their orientation when cooling in a magnetic field, and applying these formulae to the phenomena observed in the old vases he calculated the magnetic dip at the time and place of manufacture. His observations led him to infer that in Central Italy inclination was actually southerly for some centuries prior to 600B.C., when it changed sign. In 400B.C.it was about 20°N.; since 100B.C.the change has been relatively small. L. Mercanton90similarly investigated the magnetization of baked clay vases from the lake dwellings of Neuchatel, whose epoch is supposed to be from 600 to 800B.C.The results he obtained were, however, closely similar to those observed in recent vases made where the inclination was about 63°N., and he concluded in direct opposition to Folgheraiter that inclination in southern Europe has not undergone any very large change during the last 2500 years. Folgheraiter’s methods have been extended to natural rocks. Thus B. Brunhes91found several cases of clay metamorphosed by adjacent lava flows and transformed into a species of natural brick. In these cases the clay has a determinate direction of magnetization agreeing with that of the volcanic rock, so it is natural to assume that this direction coincided with that of the dip when the lava flow occurred. In drawing inferences, allowance must of course be made for any tilting of the strata since the volcanic outburst. From one case in France in the district of St Flour, where the volcanic action is assigned to the Miocene Age, Brunhes inferred a southerly dip of some 75°. Until a variety of cases have been critically dealt with, a suspension of judgment is advisable, but if the method should establish its claims to reliability it obviously may prove of importance to geology as well as to terrestrial magnetism.§ 55. Magnetic phenomena in the polar regions have received considerable attention of late years, and the observed results are of so exceptional a character as to merit separate consideration. One feature, the large amplitude of the regular diurnalPolar Phenomena.inequality, is already illustrated by the data for Jan Mayen and South Victoria Land in Tables VIII. to XI. In the case, however, of declination allowance must be made for the small size of H. If a force F perpendicular to the magnetic meridian causes a change ΔD in D then ΔD = F/H. Thus at the “Discovery’s” winter quarters in South Victoria Land, where the value of H is only about 0.36 of that at Kew, a change of 45′ in D would be produced by a force which at Kew would produce a change of only 16′. Another feature, which, however, may not be equally general, is illustrated by the data for Fort Rae and South Victoria Land in Table XVII. It will be noticed that it is the 24-hour term in the Fourier analysis of the regular diurnal inequality which is specially enhanced. The station in South Victoria Land—the winter quarters of the “Discovery” in 1902-1904—was at 77° 51′ S. lat.; thus the sun did not set from November to February (midsummer), nor rise from May to July (midwinter). It might not thus have been surprising if there had been an outstandingly large seasonal variation in the type of the diurnal inequality. As a matter of fact, however, the type of the inequality showed exceptionally small variation with the season, and the amplitude remained large throughout the whole year. Thus, forming diurnal inequalities for the three midsummer months and for the three midwinter months, we obtain the following amplitudes for the range of the several elements92:—D.H.V.I.Midsummer64′.157γ58γ2′.87Midwinter26′.825γ18γ1′.23The most outstanding phenomenon in high latitudes is the frequency and large size of the disturbances. At Kew, as we saw in § 25, the absolute range in D exceeds 20′ on only 12% of the total number of days. But at the “Discovery’s” winter quarters, about sun-spot minimum, the range exceeded 1° on 70%, 2° on 37%, and 3° on fully 15% of the total number of days. One day in 25 had a range exceeding 4°. During the three midsummer months, only one day out of 111 had a range under 1°, and even at midwinter only one day in eight had a range as small as 30′. The H range at the “Discovery’s” station exceeded 100γ on 40% of the days, and the V range exceeded 100γ on 32% of the days.The special tendency to disturbance seen in equinoctial months in temperate latitudes did not appear in the “Discovery’s” records in the Antarctic. D ranges exceeding 3° occurred on 11% of equinoctial days, but on 40% of midsummer days. The preponderance of large movements at midsummer was equally apparent in the other elements. Thus the percentage of days having a V range over 200γ was 21 at midsummer, as against 3 in the four equinoctial months.At the “Discovery’s” station small oscillations of a few minutes’ duration were hardly ever absent, but the character of the larger disturbances showed a marked variation throughout the 24 hours.Those of a very rapid oscillatory character were especially numerous in the morning between 4 and 9 a.m. In the late afternoon and evening disturbances of a more regular type became prominent, especially in the winter months. In particular there were numerous occurrences of a remarkably regular type of disturbance, half the total number of cases taking place between 7 and 9 p.m. This “special type of disturbance” was divisible into two phases, each lasting on the average about 20 minutes. During the first phase all the elements diminished in value, during the second phase they increased. In the case of D and H the rise and fall were about equal, but the rise in V was about 3½ times the preceding fall. The disturbing force—on the north pole—to which the first phase might be attributed was inclined on the average about 5°½ below the horizon, the horizontal projection of its line of action being inclined about 41°½ to the north of east. The amplitude and duration of the disturbances of the “special type” varied a good deal; in several cases the disturbing force considerably exceeded 200γ. A somewhat similar type of disturbance was observed by Kr. Birkeland93at Arctic stations also in 1902-1903, and was called by him the “polar elementary” storm. Birkeland’s record of disturbances extends only from October 1902 to March 1903, so it is uncertain whether “polar elementary” storms occur during the Arctic summer. Their usual time of occurrence seems to be the evening. During their occurrence Birkeland found that there was often a great difference in amplitude and character between the disturbances observed at places so comparatively near together as Iceland, Nova Zembla and Spitzbergen. This led him to assign the cause to electric currents in the Arctic, at heights not exceeding a few hundred kilometres, and he inferred from the way in which the phenomena developed that the seat of the disturbances often moved westward, as if related in some way to the sun’s position. Contemporaneously with the “elementary polar” storms in the Arctic Birkeland found smaller but distinct movements at stations all over Europe; these could generally be traced as far as Bombay and Batavia, and sometimes as far as Christchurch, New Zealand. Chree,92on the other hand, working up the 1902-1904 Antarctic records, discovered that during the larger disturbances of the “special type” corresponding but much smaller movements were visible at Christchurch, Mauritius, Kolaba, and even at Kew. He also found that in the great majority of cases the Antarctic curves were specially disturbed during the times of Birkeland’s “elementary polar” storms, the disturbances in the Arctic and Antarctic being of the same order of magnitude, though apparently of considerably different type.Examining the more prominent of the sudden commencements of magnetic disturbances in 1902-1903 visible simultaneously in the curves from Kew, Kolaba, Mauritius and Christchurch, Chree found that these were all represented in the Antarctic curves by movements of a considerably larger size and of an oscillatory character. In a number of cases Birkeland observed small simultaneous movements in the curves of his co-operating stations, which appeared to be at least sometimes decidedly larger in the equatorial than the northern temperate stations. These he described as “equatorial” perturbations, ascribing them to electric currents in or near the plane of the earth’s magnetic equator, at heights of the order of the earth’s radius. It was found, however, by Chree that in many, if not all, of these cases there were synchronous movements in the Antarctic, similar in type to those which occurred simultaneously with the sudden commencements of magnetic storms, and that these Antarctic movements were considerably larger than those described by Birkeland at the equatorial stations. This result tends of course to suggest a somewhat different explanation from Birkeland’s. But until our knowledge of facts has received considerable additions all explanations must be of a somewhat hypothetical character.In 1831 Sir James Ross94observed a dip of 89° 59′ at 70° 5′ N., 96° 46′ W., and this has been accepted as practically the position of the north magnetic pole at the time. The position of the south magnetic pole in 1840 as deduced from theMagnetic Poles.Antarctic observations made by the “Erebus” and “Terror” expedition is shown in Sabine’s chart as about 73° 30′ S., 147° 30′ E. In the more recent chart in J. C. Adams’sCollected Papers, vol. 2, the position is shown as about 73° 40′ S., 147° 7′ E. Of late years positions have been obtained for the south magnetic pole by the “Southern Cross” expedition of 1898-1900 (A), by the “Discovery” in 1902-1904 (B), and by Sir E. Shackleton’s expedition 1908-1909 (C). These are as follow:(A) 72° 40′ S., 152° 30′ E.(B) 72° 51′ S., 156° 25′ E.(C) 72° 25′ S., 155° 16′ E.Unless the diurnal inequality vanishes in its neighbourhood, a somewhat improbable contingency considering the large range at the “Discovery’s” winter quarters, the position of the south magnetic pole has probably a diurnal oscillation, with an average amplitude of several miles, and there is not unlikely a larger annual oscillation. Thus even apart from secular change, no single spot of the earth’s surface can probably claim to be a magnetic pole in the sense popularly ascribed to the term. If the diurnal motion were absolutely regular, and carried the point where the needle is vertical round a closed curve, the centroid of that curve—though a spot where the needle is never absolutely vertical—would seem to have the best claim to the title. It should also be remembered that when the dip is nearly 90° there are special observational difficulties. There are thus various reasons for allowing a considerable uncertainty in positions assigned to the magnetic poles. Conclusions as to change of position of the south magnetic pole during the last ten years based on the more recent results (A), (B) and (C) would, for instance, possess a very doubtful value. The difference, however, between these recent positions and that deduced from the observations of 1840-1841 is more substantial, and there is at least a moderate probability that a considerable movement towards the north-east has taken place during the last seventy years.See publications of individual magnetic observatories, more especially the Russian (Annales de l’Observatoire Physique Central), the French (Annales du Bureau Central Météorologique de France), and those of Kew, Greenwich, Falmouth, Stonyhurst, Potsdam, Wilhelmshaven, de Bilt, Uccle, O’Gyalla, Prague, Pola, Coimbra, San Fernando, Capo di Monte, Tiflis, Kolaba, Zi-ka-wei, Hong-Kong, Manila, Batavia, Mauritius, Agincourt (Toronto), the observatories of the U.S. Coast and Geodetic Survey, Rio de Janeiro, Melbourne.In the references below the following abbreviations are used: B.A. =British Association Reports; Batavia =Observations made at the Royal ... Observatory at Batavia; M.Z. =Meteorologische Zeitschrift, edited by J. Hann and G. Hellman; P.R.S. =Proceedings of the Royal Society of London; P.T. =Philosophical Transactions; R. =Repertorium für Meteorologie, St Petersburg; T.M. =Terrestrial Magnetism, edited by L. A. Bauer; R.A.S. Notices =Monthly Notices of the Royal Astronomical Society. Treatises are referred to by the numbers attached to them;e.g.(1) p. 100 means p. 100 of Walker’sTerrestrial Magnetism.
The question whether lunar influence increases with sun-spot frequency is obviously of considerable theoretical interest. Balfour Stewart in the 9th edition of this encyclopaedia gave some data indicating an appreciably enhanced lunar influence at Trivandrum during years of sun-spot maximum, but he hesitated to accept the result as finally proved. Figee recently investigated this point at Batavia, but with inconclusive results. Attempts have also been made to ascertain how lunar influence depends on the moon’s declination and phase, and on her distance from the earth. The difficulty in these investigations is that we are dealing with a small effect, and a very long series of data would be required satisfactorily to eliminate other periodic influences.
§ 41. From an analysis of seventeen years data at St Petersburg and Pavlovsk, Leyst60concluded that all the principal planets sensibly influence the earth’s magnetism. According to his figures, all the planets except Mercury—whose influencePlanetary Influence.he found opposite to that of the others—when nearest the earth tended to deflect the declination magnet at St Petersburg to the west, and also increased the range of the diurnal inequality of declination, the latter effect being the more conspicuous. Schuster,61who has considered the evidence advanced by Leyst from the mathematical standpoint, considers it to be inconclusive.
§ 42. The best way of carrying out a magnetic survey depends on where it has to be made and on the object in view. The object that probably still comes first in importance is a knowledge of the declination, of sufficient accuracy for navigationMagnetic Surveys.in all navigable waters. One might thus infer that magnetic surveys consist mainly of observations at sea. This cannot however be said to be true of the past, whatever it may be of the future, and this for several reasons. Observations at sea entail the use of a ship, specially constructed so as to be free from disturbing influence, and so are inherently costly; they are also apt to be of inferior accuracy. It might be possible in quiet weather, in a large vessel free from vibration, to observe with instruments of the highest precision such as a unifilar magnetometer, but in the ordinary surveying ship apparatus of less sensitiveness has to be employed. The declination is usually determined with some form of compass. The other elements most usually found directly at sea are the inclination and the total force, the instrument employed being a special form of inclinometer, such as the Fox circle, which was largely used by Ross in the Antarctic, or in recent years the Lloyd-Creak. This latter instrument differs from the ordinary dip-circle fitted for total force observations after H. Lloyd’s method mainly in that the needles rest in pivots instead of on agate edges. To overcome friction a projecting pin on the framework is scratched with a roughened ivory plate.
The most notable recent example of observations at sea is afforded by the cruises of the surveying ships “Galilee” and “Carnegie” under the auspices of the Carnegie Institution of Washington, which includes in its magnetic programme a general survey. To see where the ordinary land survey assists navigation, let us take the case of a country with a long seaboard. If observations were taken every few miles along the coast results might be obtained adequate for the ordinary wants of coasting steamers, but it would be difficult to infer what the declination would be 50 or even 20 miles off shore at any particular place. If, however, the land area itself is carefully surveyed, one knows the trend of the lines of equal declination, and can usually extend them with considerable accuracy many miles out to sea. One also can tell what places if any on the coast suffer from local disturbances, and thus decide on the necessity of special observations. This is by no means the only immediately useful purpose which is or may be served by magnetic surveys on land. In Scandinavia use has been made of magnetic observations in prospecting for iron ore. There are also various geological and geodetic problems to whose solution magnetic surveys may afford valuable guidance. Among the most important recent surveys may be mentioned those of the British Isles by A. Rücker and T. E. Thorpe,62of France and Algeria by Moureaux,63of Italy by Chistoni and Palazzo,64of the Netherlands by Van Ryckevorsel,65of South Sweden by Carlheim Gyllenskiöld,66of Austria-Hungary by Liznar,67of Japan by Tanakadate,68of the East Indies by Van Bemmelen, and South Africa by J. C. Beattie. A survey of the United States has been proceeding for a good many years, and many results have appeared in the publications of the U.S. Coast and Geodetic Survey, especially Bauer’sMagnetic Tables and Magnetic Charts, 1908. Additions to our knowledge may also be expected from surveys of India, Egypt and New Zealand.
For the satisfactory execution of a land survey, the observers must have absolute instruments such as the unifilar magnetometer and dip circle, suitable for the accurate determination of the magnetic elements, and they must be able to fix the exact positions of the spots where observations are taken. If, as usual, the survey occupies several years, what is wanted is the value of the elements not at the actual time of observation, but at some fixed epoch, possibly some years earlier or later. At a magnetic observatory, with standardized records, the difference between the values of a magnetic element at any two specified instants can be derived from the magnetic curves. But at an ordinary survey station, at a distance from an observatory, the information is not immediately available. Ordinarily the reduction to a fixed epoch is done in at least two stages, a correction being applied for secular change, and a second for the departure from the mean value for the day due to the regular diurnal inequality and to disturbance.
The reduction to a fixed epoch is at once more easy and more accurate if the area surveyed contains, or has close to its borders, a well distributed series of magnetic observatories, whose records are comparable and trustworthy. Throughout an area of the size of France or Germany, the secular change between any two specified dates can ordinarily be expressed with sufficient accuracy by a formula of the type
δ = δ0+ a (l − l0) + b (λ − λ0) . . (i),
where δ denotes secular change, l latitude and λ longitude, the letters with suffix0relating to some convenient central position. The constants δ0,a,bare to be determined from the observed secular changes at the fixed observatories whose geographical co-ordinates are accurately known. Unfortunately, as a rule, fixed observatories are few in number and not well distributed for survey purposes; thus the secular change over part at least of the area has usually to be found by repeating the observations after some years at several of the field stations. The success attending this depends on theexactitude with which the sites can be recovered, on the accuracy of the observations, and on the success with which allowance is made for diurnal changes, regular and irregular. It is thus desirable that the observations at repeat stations should be taken at hours when the regular diurnal changes are slow, and that they should not be accepted unless taken on days that prove to be magnetically quiet. Unless the secular change is exceptionally rapid, it will usually be most convenient in practice to calculate it from or to the middle of the month, and then to allow for the difference between the mean value for the month and the value at the actual hour of observation. There is here a difficulty, inasmuch as the latter part of the correction depends on the diurnal inequality, and so on the local time of the station. No altogether satisfactory method of surmounting this difficulty has yet been proposed. Rücker and Thorpe in their British survey assumed that the divergence from the mean value at any hour at any station might be regarded as made up of a regular diurnal inequality, identical with that at Kew when both were referred tolocaltime, and of a disturbance element identical with that existing at the same absolute time at Kew. Suppose, for instance, that at hourhG.M.T. the departure at Kew from the mean value for the month isd, then the corresponding departure from the mean at a station λ degrees west of Kew isd − e, whereeis the increase in the element at Kew due to the regular diurnal inequality between hourh− λ/15 and hourh. This procedure is simple, but is exposed to various criticisms. If we define a diurnal inequality as the result obtained by combining hourly readings from all the days of a month, we can assign a definite meaning to the diurnal inequality for a particular month of a particular year, and after the curves have been measured we can give exact numerical figures answering to this definition. But the diurnal inequality thus obtained differs, as has been pointed out, from that derived from a limited number of the quietest days of the month, not merely in amplitude but in phase, and the view that the diurnal changes on any individual day can be regarded as made up of a regular diurnal inequality of definite character and of a disturbance element is an hypothesis which is likely at times to be considerably wide of the mark. The extent of the error involved in assuming the regular diurnal inequality the same in the north of Scotland, or the west of Ireland, as in the south-east of England remains to be ascertained. As to the disturbance element, even if the disturbing force were of given magnitude and direction all over the British Isles—which we now know is often very far from the case—its effects would necessarily vary very sensibly owing to the considerable variation in the direction and intensity of the local undisturbed force. If observations were confined to hours at which the regular diurnal changes are slow, and only those taken on days of little or no disturbance were utilized, corrections combining the effects of regular and irregular diurnal changes could be derived from the records of fixed observations, supposed suitably situated, combined in formulae of the same type as (i).
§ 43. The field results having been reduced to a fixed epoch, it remains to combine them in ways likely to be useful. In most cases the results are embodied in charts, usually of at least two kinds, one set showing only general features, the other the chief local peculiarities. Charts of the first kind resemble the world charts (figs. 1 to 4) in being free from sharp twistings and convolutions. In these the declination for instance at a fixed geographical position on a particular isogonal is to be regarded as really a mean from a considerable surrounding area.
Various ways have been utilized for arriving at theseterrestrial isomagnetics—as Rücker and Thorpe call them—of which an elaborate discussion has been made by E. Mathias.69From a theoretical standpoint the simplest method is perhaps that employed by Liznar for Austria-Hungary. Letland λ represent latitude and longitude relative to a certain central station in the area. Then assume that throughout the area the value E of any particular magnetic element is given by a formula
E = E0+ al + bλ + cl² + dλ² + elλ,
where E0, a, b, c, d, e are absolute constants to be determined from the observations. When determining the constants, we write for E in the equation the observed value of the element (corrected for secular change, &c.) at each station, and for l and λ the latitude and longitude of the station relative to the central station. Thus each station contributes an equation to assist in determining the six constants. They can thus be found by least squares or some simpler method. In Liznar’s case there were 195 stations, so that the labour of applying least squares would be considerable. This is one objection to the method. A second is that it may allow undesirably large weight to a few highly disturbed stations. In the case of the British Isles, Rücker and Thorpe employed a different method. The area was split up intodistricts. For each district a mean was formed of the observed values of each element, and the mean was assigned to an imaginary central station, whose geographical co-ordinates represented the mean of the geographical co-ordinates of the actual stations. Want of uniformity in the distribution of the stations may be allowed for by weighting the results. Supposing E0the value of the element found for the central station of a district, it was assumed that the value E at any actual station whose latitude and longitude exceeded those of the central station by l and λ was given by E = E0+ al + bλ, with a and b constants throughout the district. Having found E0, a and b, Rücker and Thorpe calculated values of the element for points defined by whole degrees of longitude (from Greenwich) and half degrees of latitude. Near the common border of two districts there would be two calculated values, of which the arithmetic mean was accepted.
The next step was to determine by interpolation where isogonals—or other isomagnetic lines—cut successive lines of latitude. The curves formed by joining these successive points of intersection were calleddistrictlines or curves. Rücker and Thorpe’s next step was to obtain formulae by trial, giving smooth curves of continuous curvature—terrestrial isomagnetics—approximating as closely as possible to the district lines. The curves thus obtained had somewhat complicated formulae. For instance, the isogonals south of 54°.5 latitude were given for the epoch Jan. 1, 1891 by
D = 18° 37′ + 18′.5 (l − 49.5) − 3′.5 cos {45° (l − 49.5) }+ {26′.3 + 1′.5 (l − 49.5) } (λ − 4) + 0′.01 (λ − 4)² (l − 54.5)²,
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. Baillaud69have 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 Mathias69of 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 disturbancesLocal 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 termedridge lines. The corresponding loci where the largest negative values occur were calledvalley 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 Moureaux70and by E. Leyst.71In 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.72The 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.73Within 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. Field74—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.75Again, Eschenhagen76detected 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 Passalsky72at 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. Claxton77found 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°1⁄3in 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 surfaceGaussian 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
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(gmncos mλ + hmnsin mλ) ]+ Σ (r / R)n[Hmn(gm−ncos mλ + hm−nsin mλ) ],
where Σ denotes summation of m from 0 to n, followed by summation of n from 0 to ∞. In this equation gmn, &c. are constants, those with positive suffixes being what are generally termedGaussian 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 nthorder is a spherical harmonic of the nthdegree. If R be taken as unit length, as is not infrequent, the first order terms are given by
V1= r−2[g10sin l + (g11cos λ + h11sin λ) cos l].
The earth is in reality a spheroid, and in his elaborate work on the subject J. C. Adams78develops 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
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 gmn, &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.
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.79
§ 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:—
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 g20in 1842-1845 were -.00127 and -.00057, and those obtained for h31in 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, gmnmust be replaced by gmn+ gm−n, and in any equation determining Z by gmn− {n/(n + 1)} gmn; similar remarks apply to hmnand 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 gmnand hmnfrom the X and Y or from the Z equations, when gm−nand hm−nare assumed zero. If the values so found differ, values can be found for gm−nand hm−nwhich will harmonize the two sets of equations. Adams gives the values obtained from the X, Y and the Z equations separately for theGaussian constants. The following are examples of the values thence deducible for the coefficients of the positive power series:—
Compared to g40, g50and g60the values here found for g−40, g−50and g−60are 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−11and 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.
§ 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
M = R3{ (g10)2+ (g11)2+ (h11)2}1/2,
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 thenormalfield—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. Schmidt80Earth-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 Bezold81and Bauer82the 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. Fritsche83treating the problem similarly, but for two epochs, 1842 and 1885, got conspicuously different results for the two epochs, Bauer84has 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 anupwardlydirected current.
Table XLVI.—Earth-air Currents, after Bauer.
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 of1⁄30ampere 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,85and to Austria by Liznar,86but with negative results. The question has also been considered in detail by Tanakadate68in discussing the magnetic survey of Japan. He makes the criticism that the taking of a line integral round theboundaryof 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 nthorder, 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. Liznar86acompared this equation with the observed results of his Austrian survey, subdividing his stations into three groups accordingto 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.68If 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.
The − sign in Table XLVII. denotes a decrease in the numerical values of H, V and I, and a diminution in westerly declination. If we except the case of the westerly component of force—not shown in the table—the accordance between the results from the two methods in the case of Japan is extraordinarily close, and there is no very marked tendency for the one method to give larger values than the other. In the case of Great Britain and Austria the differences between the two sets of calculated values though not large are systematic, the 3h/R formula invariably showing the larger reduction with altitude in both H and V. Tanakadate was so satisfied with the accordance of the two methods in Japan, that he employed his method to reduce all observed Japanese values to sea-level. At a few of the highest Japanese stations the correction thus introduced into the value of H was of some importance, but at the great majority of the stations the corrections were all insignificant.
§ 53. Schuster87has calculated a potential analogous to the Gaussian potential, from which the regular diurnal changes of the magnetic elements all over the earth may be derived. From the mean summer and winter diurnal variationsSchuster’s Diurnal Variation Potential.of the northerly and easterly components of force during 1870 at St Petersburg, Greenwich, Lisbon and Bombay, he found the values of 8 constants analogous to Gaussian constants; and from considerations as to the hours of occurrence of the maxima and minima of vertical force, he concluded that the potential, unlike the Gaussian, must proceed in positive powers ofr, and so answer to forces external to the earth. Schuster found, however, that the calculated amplitudes of the diurnal vertical force inequality did not accord well with observation; and his conclusion was that while the original cause of the diurnal variation is external, and consists probably of electric currents in the atmosphere, there are induced currents inside the earth, which increase the horizontal components of the diurnal inequality while diminishing the vertical. The problem has also been dealt with by H. Fritsche,88who concludes, in opposition to Schuster, that the forces are partly internal and partly external, the two sets being of fairly similar magnitude. Fritsche repeats the criticism (already made in the last edition of this encyclopaedia) that Schuster’s four stations were too few, and contrasts their number with the 27 from which his own data were derived. On the other hand, Schuster’s data referred to one and the same year, whereas Fritsche’s are from epochs varying from 1841 to 1896, and represent in some cases a single year’s observations, in other cases means from several years. It is clearly desirable that a fresh calculation should be made, using synchronous data from a considerable number of well distributed stations; and it should be done for at least two epochs, one representing large, the other small sun-spot frequency. The year 1870 selected by Schuster had, as it happened, a sun-spot frequency which has been exceeded only once since 1750; so that the magnetic data which he employed were far from representative of average conditions.
§ 54. It was discovered by Folgheraiter89that old vases from Etruscan and other sources are magnetic, and from combined observation and experiment he concluded that they acquired their magnetization when cooling after being baked, andMagnetization of Vases, &c.retained it unaltered. From experiments, he derived formulae connecting the magnetization shown by new clay vases with their orientation when cooling in a magnetic field, and applying these formulae to the phenomena observed in the old vases he calculated the magnetic dip at the time and place of manufacture. His observations led him to infer that in Central Italy inclination was actually southerly for some centuries prior to 600B.C., when it changed sign. In 400B.C.it was about 20°N.; since 100B.C.the change has been relatively small. L. Mercanton90similarly investigated the magnetization of baked clay vases from the lake dwellings of Neuchatel, whose epoch is supposed to be from 600 to 800B.C.The results he obtained were, however, closely similar to those observed in recent vases made where the inclination was about 63°N., and he concluded in direct opposition to Folgheraiter that inclination in southern Europe has not undergone any very large change during the last 2500 years. Folgheraiter’s methods have been extended to natural rocks. Thus B. Brunhes91found several cases of clay metamorphosed by adjacent lava flows and transformed into a species of natural brick. In these cases the clay has a determinate direction of magnetization agreeing with that of the volcanic rock, so it is natural to assume that this direction coincided with that of the dip when the lava flow occurred. In drawing inferences, allowance must of course be made for any tilting of the strata since the volcanic outburst. From one case in France in the district of St Flour, where the volcanic action is assigned to the Miocene Age, Brunhes inferred a southerly dip of some 75°. Until a variety of cases have been critically dealt with, a suspension of judgment is advisable, but if the method should establish its claims to reliability it obviously may prove of importance to geology as well as to terrestrial magnetism.
§ 55. Magnetic phenomena in the polar regions have received considerable attention of late years, and the observed results are of so exceptional a character as to merit separate consideration. One feature, the large amplitude of the regular diurnalPolar Phenomena.inequality, is already illustrated by the data for Jan Mayen and South Victoria Land in Tables VIII. to XI. In the case, however, of declination allowance must be made for the small size of H. If a force F perpendicular to the magnetic meridian causes a change ΔD in D then ΔD = F/H. Thus at the “Discovery’s” winter quarters in South Victoria Land, where the value of H is only about 0.36 of that at Kew, a change of 45′ in D would be produced by a force which at Kew would produce a change of only 16′. Another feature, which, however, may not be equally general, is illustrated by the data for Fort Rae and South Victoria Land in Table XVII. It will be noticed that it is the 24-hour term in the Fourier analysis of the regular diurnal inequality which is specially enhanced. The station in South Victoria Land—the winter quarters of the “Discovery” in 1902-1904—was at 77° 51′ S. lat.; thus the sun did not set from November to February (midsummer), nor rise from May to July (midwinter). It might not thus have been surprising if there had been an outstandingly large seasonal variation in the type of the diurnal inequality. As a matter of fact, however, the type of the inequality showed exceptionally small variation with the season, and the amplitude remained large throughout the whole year. Thus, forming diurnal inequalities for the three midsummer months and for the three midwinter months, we obtain the following amplitudes for the range of the several elements92:—
The most outstanding phenomenon in high latitudes is the frequency and large size of the disturbances. At Kew, as we saw in § 25, the absolute range in D exceeds 20′ on only 12% of the total number of days. But at the “Discovery’s” winter quarters, about sun-spot minimum, the range exceeded 1° on 70%, 2° on 37%, and 3° on fully 15% of the total number of days. One day in 25 had a range exceeding 4°. During the three midsummer months, only one day out of 111 had a range under 1°, and even at midwinter only one day in eight had a range as small as 30′. The H range at the “Discovery’s” station exceeded 100γ on 40% of the days, and the V range exceeded 100γ on 32% of the days.
The special tendency to disturbance seen in equinoctial months in temperate latitudes did not appear in the “Discovery’s” records in the Antarctic. D ranges exceeding 3° occurred on 11% of equinoctial days, but on 40% of midsummer days. The preponderance of large movements at midsummer was equally apparent in the other elements. Thus the percentage of days having a V range over 200γ was 21 at midsummer, as against 3 in the four equinoctial months.
At the “Discovery’s” station small oscillations of a few minutes’ duration were hardly ever absent, but the character of the larger disturbances showed a marked variation throughout the 24 hours.Those of a very rapid oscillatory character were especially numerous in the morning between 4 and 9 a.m. In the late afternoon and evening disturbances of a more regular type became prominent, especially in the winter months. In particular there were numerous occurrences of a remarkably regular type of disturbance, half the total number of cases taking place between 7 and 9 p.m. This “special type of disturbance” was divisible into two phases, each lasting on the average about 20 minutes. During the first phase all the elements diminished in value, during the second phase they increased. In the case of D and H the rise and fall were about equal, but the rise in V was about 3½ times the preceding fall. The disturbing force—on the north pole—to which the first phase might be attributed was inclined on the average about 5°½ below the horizon, the horizontal projection of its line of action being inclined about 41°½ to the north of east. The amplitude and duration of the disturbances of the “special type” varied a good deal; in several cases the disturbing force considerably exceeded 200γ. A somewhat similar type of disturbance was observed by Kr. Birkeland93at Arctic stations also in 1902-1903, and was called by him the “polar elementary” storm. Birkeland’s record of disturbances extends only from October 1902 to March 1903, so it is uncertain whether “polar elementary” storms occur during the Arctic summer. Their usual time of occurrence seems to be the evening. During their occurrence Birkeland found that there was often a great difference in amplitude and character between the disturbances observed at places so comparatively near together as Iceland, Nova Zembla and Spitzbergen. This led him to assign the cause to electric currents in the Arctic, at heights not exceeding a few hundred kilometres, and he inferred from the way in which the phenomena developed that the seat of the disturbances often moved westward, as if related in some way to the sun’s position. Contemporaneously with the “elementary polar” storms in the Arctic Birkeland found smaller but distinct movements at stations all over Europe; these could generally be traced as far as Bombay and Batavia, and sometimes as far as Christchurch, New Zealand. Chree,92on the other hand, working up the 1902-1904 Antarctic records, discovered that during the larger disturbances of the “special type” corresponding but much smaller movements were visible at Christchurch, Mauritius, Kolaba, and even at Kew. He also found that in the great majority of cases the Antarctic curves were specially disturbed during the times of Birkeland’s “elementary polar” storms, the disturbances in the Arctic and Antarctic being of the same order of magnitude, though apparently of considerably different type.
Examining the more prominent of the sudden commencements of magnetic disturbances in 1902-1903 visible simultaneously in the curves from Kew, Kolaba, Mauritius and Christchurch, Chree found that these were all represented in the Antarctic curves by movements of a considerably larger size and of an oscillatory character. In a number of cases Birkeland observed small simultaneous movements in the curves of his co-operating stations, which appeared to be at least sometimes decidedly larger in the equatorial than the northern temperate stations. These he described as “equatorial” perturbations, ascribing them to electric currents in or near the plane of the earth’s magnetic equator, at heights of the order of the earth’s radius. It was found, however, by Chree that in many, if not all, of these cases there were synchronous movements in the Antarctic, similar in type to those which occurred simultaneously with the sudden commencements of magnetic storms, and that these Antarctic movements were considerably larger than those described by Birkeland at the equatorial stations. This result tends of course to suggest a somewhat different explanation from Birkeland’s. But until our knowledge of facts has received considerable additions all explanations must be of a somewhat hypothetical character.
In 1831 Sir James Ross94observed a dip of 89° 59′ at 70° 5′ N., 96° 46′ W., and this has been accepted as practically the position of the north magnetic pole at the time. The position of the south magnetic pole in 1840 as deduced from theMagnetic Poles.Antarctic observations made by the “Erebus” and “Terror” expedition is shown in Sabine’s chart as about 73° 30′ S., 147° 30′ E. In the more recent chart in J. C. Adams’sCollected Papers, vol. 2, the position is shown as about 73° 40′ S., 147° 7′ E. Of late years positions have been obtained for the south magnetic pole by the “Southern Cross” expedition of 1898-1900 (A), by the “Discovery” in 1902-1904 (B), and by Sir E. Shackleton’s expedition 1908-1909 (C). These are as follow:
Unless the diurnal inequality vanishes in its neighbourhood, a somewhat improbable contingency considering the large range at the “Discovery’s” winter quarters, the position of the south magnetic pole has probably a diurnal oscillation, with an average amplitude of several miles, and there is not unlikely a larger annual oscillation. Thus even apart from secular change, no single spot of the earth’s surface can probably claim to be a magnetic pole in the sense popularly ascribed to the term. If the diurnal motion were absolutely regular, and carried the point where the needle is vertical round a closed curve, the centroid of that curve—though a spot where the needle is never absolutely vertical—would seem to have the best claim to the title. It should also be remembered that when the dip is nearly 90° there are special observational difficulties. There are thus various reasons for allowing a considerable uncertainty in positions assigned to the magnetic poles. Conclusions as to change of position of the south magnetic pole during the last ten years based on the more recent results (A), (B) and (C) would, for instance, possess a very doubtful value. The difference, however, between these recent positions and that deduced from the observations of 1840-1841 is more substantial, and there is at least a moderate probability that a considerable movement towards the north-east has taken place during the last seventy years.
See publications of individual magnetic observatories, more especially the Russian (Annales de l’Observatoire Physique Central), the French (Annales du Bureau Central Météorologique de France), and those of Kew, Greenwich, Falmouth, Stonyhurst, Potsdam, Wilhelmshaven, de Bilt, Uccle, O’Gyalla, Prague, Pola, Coimbra, San Fernando, Capo di Monte, Tiflis, Kolaba, Zi-ka-wei, Hong-Kong, Manila, Batavia, Mauritius, Agincourt (Toronto), the observatories of the U.S. Coast and Geodetic Survey, Rio de Janeiro, Melbourne.
In the references below the following abbreviations are used: B.A. =British Association Reports; Batavia =Observations made at the Royal ... Observatory at Batavia; M.Z. =Meteorologische Zeitschrift, edited by J. Hann and G. Hellman; P.R.S. =Proceedings of the Royal Society of London; P.T. =Philosophical Transactions; R. =Repertorium für Meteorologie, St Petersburg; T.M. =Terrestrial Magnetism, edited by L. A. Bauer; R.A.S. Notices =Monthly Notices of the Royal Astronomical Society. Treatises are referred to by the numbers attached to them;e.g.(1) p. 100 means p. 100 of Walker’sTerrestrial Magnetism.