2 P.M., the time of the day when the diurnal inequality is most in evidence, the value of seems to vary very little with the season. Curves whose radius vector is p, and vectorial angle, illustrate the variations of diurnal inequality in a conspicuous way. Numerous curves of this kind, illustrating the results obtained at Greenwich for the years 1841-57, were given by Sir G. B. Airy in the 'Phil. Trans.' for 1863, and a smaller number, referring apparently to the years 1873-75, were also given by Airy in the 'Phil. Trans.' for 1885. Several applications of these curves to Dublin results appear in Dr. Lloyd's Treatise.' Following these examples I have drawn curves 11, 12, and 13, Plate VI., based on the results in Table VII., for midwinter, midsummer, and the whole year. The hours are stated beside the points on the curves to which they refer. The prominent loop on the midwinter curve should be noticed. The existence of a loop on the mean winter curve and its absence in the mean summer curve for Dublin are referred to by Lloyd (loc. cit., p. 187). The tendency to loops in winter months is conspicuous in Airy's Greenwich curves. Relative Intensity of the Forces to which the Diurnal Inequality is due throughout the year. § 13. Without knowing the true nature of the disturbing force or forces to which the diurnal inequality is due, it is difficult to suggest any wholly satisfactory measure of intensity. Assuming the inequality to be a composite phenomenon, the counteraction of opposing forces may produce the same effect at one hour as the absence of forces at another. All the phenomena seem, however, to point to greatly increased activity of disturbing forces in summer. The data given above as to the mean values of p throughout the day at midwinter and midsummer make the former mean only 374 of the latter. Evidence in the same direction is supplied by the following table VIII., which gives for each month, each quarter and half-year, and for the whole year, the sum of the hourly departures from the mean for the day, irrespective of sign, along with the range, or algebraical difference between the extremes. To show how comparatively little depends on how the non-cyclic element is dealt with, the table, in addition to the results obtained after the non-cyclic element has been eliminated, gives likewise the range deduced from the uncorrected data for each month. The fractions, which the table records, refer exclusively to the corrected data. The range and the sum of the inequalities in both declination and horizontal force show a distinct minimum in December. In the declination both the range and the sum of the inequalities present an absolute maximum in August, with apparently a second, only slightly smaller maximum, in May. The variation, however, especially of the range, is so small from April to August that it would hardly be safe to conclude this was the normal phenomenon in a year of 'quiet' days. In the horizontal force there would appear to be a single maximum, whether for range or sum of inequalities, in July; but the difference between the results and those for May, June, and August is extremely small. §14. These conclusions are only partly in accord with those got out by Dr. Balfour Stewart for a long series of years, 1858-73, at Kew, and by Mr. W. Ellis,2 for a still longer series of years, 1841-77, at Greenwich. 1 Proc. Roy. Soc., vol. xxvi. 1877, p. 103. 2 Phil. Trans. for 1880, p. 544. The tables by Dr. Stewart and Mr. Ellis both give April as the month in which occurs the largest maximum in the declination range. Dr. Stewart's table gives slightly smaller, and nearly equal, maxima in June and August. In Mr. Ellis's table the mean range for August is decidedly higher than the mean for June, which latter, however, is a shade higher than the mean for May and decidedly higher than the mean for July. Again, the difference between the maximum and minimum ranges of declination in Table VIII. is much greater than in either of the other tables referred to. Thus, by Table VIII., the corrected range varies from 12'-23 to 3'97, while in the general mean of the monthly results from the thirty-seven years included in Mr. Ellis's table the extremes are 11'96 and 5'78, and in Dr. Stewart's table the ratio of the maximum to the minimum monthly range is only 2 : 1. Dr. Stewart's table is confined to declination, but Mr. Ellis gives results for the range of horizontal force as well. The general means he gives for the four months, April to July, are very nearly equal; the greatest value, that for April, being about 2 per cent. only in excess of the least of the four, that for May. The difference between the greatest and least of the mean monthly ranges of horizontal force in Mr. Ellis's table is again much less than the corresponding difference in Table VIII., the minimum, appearing in December, bearing to the April maximum the ratio 42: 100. The most conspicuous difference between Table VIII. and the results of Dr. Stewart and Mr. Ellis is that it shows no trace of a maximum of activity in April, and indicates markedly increased activity in August. It would, however, be precipitate to assume that the time of maximum Corrected Fraction of mean for year activity falls later in the year now than in the epochs dealt with by Dr. Stewart and Mr. Ellis. The different character of the materials employed in the several cases must be borne in mind. Dr. Stewart took into account all observations except the comparatively small percentage falling under General Sabine's definition of disturbed, and Mr. Ellis took all but days of considerable magnetic disturbance; thus the material dealt with by both possessed presumably much greater heterogeneousness than that dealt with in Table VIII. Again, the mode in which the results were treated was different. Dr. Stewart, it is true, apparently took the means from groups of four successive years, but Mr. Ellis took the mean of the ranges deduced from each single year's records. The taking a group of years probably depresses the mean range most in those months in which the progress of the diurnal inequality is least regular. Harmonic Analysis of Monthly Ranges. § 15. I have analysed into harmonic terms the quantities measuring mean range for month the departures from unity of the fractions for both mean range for year declination and H.F. Putting for brevity where is time in months from the middle of January, I find for the declination mean range for month mean range for year 140 cos x+14 sin x-11 cos 2x +08 sin 2x+01 cos 3x-04 sin3x+01 cos 4x+00 sin 4x +04 cos 5x-·03 sin 5x+03 cos 6x for the Horizontal Force (9) mean range for month -1=-43 cos x+08 sin x- 07 cos 2x mean range for year +03 sin 2x+00 cos 3x-00 sin 3x+03 cos 4x+03 sin 4x +03 cos 5x+01 sin 5x+01 cos 6x (10) In both cases the terms whose period is the full year are greatly predominant. There appears also in both cases an appreciable semi-annual variation; but the terms with shorter periods are very small, and little weight can be attached to their numerical measure. Annual Variation. § 16. The variation of a magnetic element throughout the year, like its variation throughout a quiet day, is most simply regarded as composed of two parts, a uniform drift or secular variation, and a cyclic portion, which may be called the annual inequality. This is a somewhat arbitrary separation. The increase of the horizontal force, for instance, from year to year, got out from the observations of most observatories, is far from uniform; and if this be a true phenomenon the hypothesis that the secular drift is uniform throughout the whole of one year can hardly claim a physical basis. It is, however, at any rate a convenient mathematical fiction whose adoption can do no harm when its true character is explicitly recognised. The following data are extracted from the annual Kew 'Reports : The mean values thence deduced for the secular variation are a decrease annually of 6'9 in declination, and an increase annually of 10-6 x 195 C.G.S. units in horizontal force. It will be assumed that these secular variations proceed uniformly throughout the mean year, got by combining the five years 1890-94. To eliminate the secular variation one adds to the observed values +0.575t in the case of the declination, and 10-6 x 1.625t in the case of the horizontal force, t being the time elapsed in months from the middle of the mean year. Subtracting the mean value for the year from the monthly means thus corrected, one obtains the annual inequality. § 17. To make the true position of affairs clear, a brief explanation is necessary of how the magnetic curves are standardised at Kew. Of late years the practice has been to determine the value of the zero line for each month's curves by reference solely to the absolute observations of that month. The same instruments are used for each observation of an element, and every observation is independent of the others, except that the constant usually called P in the formula for the horizontal forcei.e., the coefficient of the secondary term in the expression for the deflecting force-is determined from a whole year's observations. One of the most probable and subtle causes of error one has to provide against in getting out an annual variation of any physical phenomenon is a possible secular or annual variation in the measuring instruments. This is especially the case with apparatus sensitive to changes of temperature. Now it is unquestionably true that the horizontal force magnetograph is affected by changes of temperature, and though the underground chamber it is worked in at Kew has a very small diurnal variation of temperature, it has a considerable annual fluctuation. Thus, however carefully temperature corrections might be determined and applied, the suspicion of an annual inequality' being far other than it seemed might not unreasonably be entertained, if the ultimate reference were to the magnetograph curves, either unstandardised, or standardised by reference to the mean of a year's absolute observations. It is partly to provide against this that each month's curves are referred to that one month's absolute observations. These observations are taken about once a week and scattered over the month, so that any secular or annual variation in the magnetographs themselves must be very nearly eliminated. As regards the absolute instruments there is, so to speak, no higher court of appeal. There is no obvious ground for suspicion. The horizontal force magnet has a temperature correction to apply, but this is only to allow for the difference in its temperatures at the times of the vibration and deflection experiments in the same observation. This difference in temperature is very small and very irregular, and even if the tempera |