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exactitude 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 to local time, and of a disturbance element identical with that existing at the same absolute time at Kew. Suppose, for instance, that at hour h G.M.T. the departure at Kew from the mean value for the month is d, then the corresponding departure from the mean at a station A degrees west of Kew is rl-e, where e is the increase in the element at Kew due to the regular diurnal ine uality between hour h-A/I5 and hour h. This procedure is simp(le, 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 intens ity 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. I to 4) in being free from sharp twistin s 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 these terrestrial isomagnetics-as Rucker and Thorpe call them-of which an elaborate discussion has been made by E. Mathias.” From a theoretical standpoint the simplest method is perhaps that employed by Liznar for Austria-Hungary. Let l and A 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 =E., +al+b>.+¢r2+.1>+el>~,

where Ea, a, Zz, 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 A 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, Rucker and Thorpe employed a different method. The area was split up into districts. 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 E0 the 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) was given by E=E0-I-al-l-b), with a and b constants throughout the TERRESTRIAL

district. Having found EO, a and b, Rucker 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 called district lines or curves. Rucker 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=1§ ° 37'p“18'-50-49-5>-3'°5, c<>s l45°(l-49~5>} +l26 °3~l-I ~5<l-49'5>}<>~-4>+0 -01(>-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 ]an. 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. Baillaud69 have applied to Rucker and Thorpe's observations a method which is a combination of Rucker and Thorpe's and of Liznar's. Taking Rucker 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. I, 1891, was D =2o° 45'-89+-53474>+-347161-I—oooo21>

-1- -ooo3431> - '00023Ql2,

where l-I-(53° 30'-5) represents the latitude, and (A--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 Rucker 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 Rucker and Thorpe from their own formulae. The mean numerical (calculated ~ observed) differences from the two different methods are almost exactly the same-being approximately IO' for declination, 5% for inclination, and 70y for horizontal force. The applications by Mathias” of his method to the survey data of France obtained by Moureaux, and those of the Netherlands obtained by van Rijckevorsel, 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; Rucker and Thorpe regarded their terrestrial isomagnetics and the corresponding formulae as representing the normal field that would exist in the absence of disturbances peculiar to the neighbourhood. Subtracting the forces L°°aIms derived from the formulae from those observed, we turbances 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, Rucker and Thorpe regard it as positive, and the loci where the largest ositive values occur they termed ridge lines. The corresponding l)oci where the largest negative values occur were called valley lines. In the British Isles Rucker 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. Riicker 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, of Kursk in Russia described by

notably those in the province

Moureaux 70 and by E. Leyst.” In Kursk Leyst observed declinations varying from 0° to 6o°,

£5 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