Table XV.-Diurnal Inequalities in N. and W. at Falmouth (unit ry). months of one year, or for all the Januarys of a series of years, we have only to take their arithmetic means our. 1 2 3 4 5 6 7 S 9 10 II 12 to obtain the corresponding constants for the mea n f diurnal inequality of the year, or for the diurnal in- a.m. + 6 + 5 + 5 + 5 + 6 + 6 + 5 + I - 6 -14 -20 -20 equality of the average January of the series of years. p.m. -17 -12 - 6 - 1 + 3 + 6 + 9 + 9 + 9 + 8 + 7 + 7 This, however, is obviously not true of the c or a constants, unless the phase angle is absolutely unchanged W § a.m. - 2 - 2 - 3 - 4 - 6 - 9 -13 -17 -19 -13 - 3 +11 throughout the contributory months or years. This is a (p.m. +20 +22 +17 +11 + 6 + 4 + 2 + 1 0 - 1 - 2 - 2 point re uiring careful attention, because when giving values of c and o. for the whole year some authorities curves, is characteristic of the winter months at Kew and Falmouth. The shape is less variable in summer than in winter; but even in summer the portion answering to the hours 6 p.m. to 6 a.m. varies a good deal. The object of presenting the Kew and Falmouth curves side by side is to emphasize the close resemblance between the magnetic phenomena at places in similar latitudes, though over 200 miles apart and exhibiting widely different ranges for their meteorological elements. With considerable change of latitude however the shape of vector diagrams changes largely. § 2O. Any diurnal inequality can be analysed into a series of Four, ” harmonic terms whose periods are 24 hours and sub-Se, , es multiples thereof. The series may be expressed in either of the equivalent forms:-

give the arithmetic mean of the c's and a's calculated from the diurnal inequalities of the individual months of the year, others give the values obtained for c and a. from the inean diurnal inequality of the whole year. The former method inevitably supplies a larger value for 6 than the latter, supposing to vary with the season. At some observatories, e.g. Greenwich and Batavia, it has long been customary to publish every year values of the Fourier coefficients for each month, and to include other elements besides the declination. For a thoroughly satisfactory comparison of different stations, it is necessary to have data from one and the same epoch; and preferably that epoch should include at least one 11-year period. There arejhowever, few stations which can supply the data required for such a comparison and we have to make the best of what is available. Information is naturally most copious for the declination. For this element E. Engelenburg 2° I. 21 23 30 39

W. 20 24 46 54

al cos t+b; sin t+a2 cos 2t+b2 sin 2t+ . . (i) cr sin (H-ar) +62 sin (2t+a.2)+ .... (ii) gives Values of Eh 62, Table XVI -Ranges in Diurnal Inequalities at Falmouth (unit Iy). ljan. Feb. Mar. April. May. June. July. !Aug. Sept. 39 37 37 39 36

55 55 54 56 51

32 24 I5

Oct. Nov. Dec.

39 24 I5

ca, ci, and of al, ag, 0.3, <14 for each month of the year for about 50 stations, ranging from Fort Rae (62° 6' N. lat.) to Cape Horn (55° 5' S. lat.). From the results for individual stations, Engelenburg derives a series of means

which he re ards as representative of II different zones of latitude. His data for individual stations refer to different epochs, and some In both forms t denotes time, counted usually from midnight, one hour of time being interpreted as 15° of angle. Form (i) is that utilized in actually calculating the constants a, b, .. . Once the a. b, . . constants are known, the c, a, . . . constants are at once derivable from the formulae:-

tan a, .=a, ./bn; c, .=a, ,/sin a., .=b, ,/cos a, .= /(a, .2-i-bf). The a, b, c, a constants are called sometimes Fourier, sometimes Bessel coefficients.

By taking a sufficient number of terms a series can always be obtained which will represent any set of diurnal inequality figures; but unless one can obtain a close approach to the observational are based on only one year's observations. The original observations also differ in reliability; thus the results are of somewhat unequal value. The mean results for Engelenburg's zones must naturally have some of the sources of uncertainty reduced; but then the fundamental idea represented by the arrangement in zones is open to question. The majority of the data in Table XVII. are taken from Engelenburg, but the phase angles have been altered so as to apply to westerly declination. The stations are arranged in order of latitude from north to south; in a few instances results are given for quiet days. The figures represent in all cases arithmetic means derived from the 12 monthly values. In the table, so far as is known, the local mean time of the observatory has been employed. This is a point requiring attention, because most observatories June. July.

18

18 N 2, M N

o

/Jf, § (2l o 4/mf;, 3

18 + 3 ' B L8. .x 0 6

x-§ >< ' 53

W F E '-rc WN - E ai

is V '5

W . E + W A g

x +

15 i 15

9 . 9 *9

9

I2, ; +f V 12 s/

is 1* 12 *

(From Phil. Trans.) FIG 7

figures from the terms possessing the periods 24, I2, 8 and 6 hours the physical significance and general utility of the analysis is somewhat problematical. In the case of the magnetic elements, the 24 and 12 hour terms are usually much the more important; the 24-hour term is generally, but by no means always, the larger of the two. The c constants give the amplitudes of the harmonic terms or waves, the a constants the phase angles. An advance of 1 hour in the time of occurrence of the first (and subsequent, if any) maximum and minimum answers to an increase of 15° in a., of 30° in az, of 45° in as, of 60° in a4 and so on. In the case of magnetic elements the phase angles not infrequently possess a somewhat large annual variation. It is thus essential libr a minute study of the phenomena at any station to carry out the analysis for the different seasons of the year, and preferably for the individual months. If the a and 17 constants are known for all the individual employ Greenwich time, or time based on Greenwich or some other national observatory, and any departure from local time enters into the values of the constants. The data for Victoria Land refer to the “ Discovery's ” 1902-1903 winter quarters, where the declination, taken westerly, was about 2o7°- 5.

As an example of the significance of the phase angles in Table XVII., take the ordinary day data for Kew. The times of occurrence of the maxima are given by l+234°=450° for the 24-hour term, 2t+39°-7=9o° or =450° for the 12-hour term, and so on, taking an hour in t as equivalent to 15°. Thus the times of the maxima are:-

24-hour term, 2 h. 24 m. p.m.; 12-hour term, I h 4I m. a.m. and p.m.

8-hour term, 4 h. 4I m. a.m., o h. 4I m. p.m., and 8 h. 41 m. p.m. 6-hour term, 0 h. 33 m. a.m. and p.m., and 6 h. 33 m. a.m. and p.m.