^{page 563 note *} Analogie de l'Électricité et du Magnétisme, t. iii p. 85. This volume contains the “Dissertation sur les mouvements irreguliers de l'Aiguille aimantée, par J. H. Van Swinden, La Haye, 1785.”

^{page 564 note *} Analogie de l'Élect. et du Mag. t. iii. p. 129. This result of Van Swinden's long and persevering labours seems to have been lost sight of.

^{page 564 note †} Œuvres de F. Arago, t. iv. p. 501, Paris, 1854.

^{page 564 note ‡} Resultate des Mag. Vereins, 1836, S. 54.

^{page 564 note §} Resultate des Mag. Obs. in München, 1846, S. 31.

^{page 564 note ║} Poggendorff's Annalen, B. 84, S. 572, Dec. 1851.

^{page 565 note *} On the Periodical Laws, &c., received Mar. 18; read May 6, 1852.—Phil. Trans., p. 183, 1852.

^{page 565 note †} Berner Mittheilungen, No. 245, 1852.

^{page 565 note ‡} Bibliotheque Universelle, Juillet et Août, 1852.

^{page 565 note §} Ueber die zehnjährige Periode; Sitz. b. der k. Akad. z. München, 1862, Bd. ii. Heft 2.

^{page 566 note *} This method was employed by me previously for the daily and monthly means of horizontal magnetic force.—Trans. Roy. Soc. Edin., vol. xxii. plates xxv., xxvi, and xxviii.

^{page 566 note †} I have been able, before printing, to add the means for several months later than those in my possession when this paper was written.

^{page 567 note *} De la déclinaison et des variations de l'Aguille aimantée, Paris, 1791.

^{page 567 note †} Phil. Trans. 1806, p. 416.

^{page 567 note ‡} Thomson's Annals of Philosophy.

^{page 567 note §} Œuvres de F. Arago, t. iv p. 501.

^{page 567 note ║} Resultate des Mag. Vereins im Jahre 1836, 1839.

^{page 567 note ¶} Resultate d. Mag. Beob. 1840–42; Result, d. Mag. Observatonums in München, 1843–45; Poggendorff Annalen, B. 84. In obtaining the yearly mean range, 1′·1 has been added to the yearly means from 1841.1 (Feb. 1), to 1842·0, derived from two-hourly observations, to reduce to the means from 1842·0 to 1845·5 from hourly observations, after 1845·5, 0′·6 is added. The object of these corrections has been to obtain comparative results from different places for the epochs of minimum (37).

^{page 567 note **} Observations made at the Mag. and Met. Obs. Dublin, vol. i. p. 89; vol. ii. p. 53. The ranges are taken from observations at 7 a.m. 1 p.m. or p.m. in 1840–43, and from the three-hourly observations after 7 a.m. in the following years.

^{page 567 note ††} Observations in Magnetism and Meteorology (Trans. Roy. Socy. Edin. vol. xvii., xviii., and xix.), 1′·2 has been added to the yearly mean ranges from January 1, 1843,to June 1, 1844.

^{page 568 note *} Observations made at the Magnetical and Meteorological Observatory at Toronto, in Canada, edited by Lt. Col. E. Sabine, vol. i. p. xii; vol. ii. p. 10. 0'·5 has been added to the yearly mean ranges from July 1, 1841, to June 1, 1842, derived from two-hourly observations, to reduce to hourly observations made afterwards.

^{page 568 note †} Observations, &c., at Hobarton, edited by Lt. Col. E. Sabine, vol. i. p. xxvi.

^{page 568 note ‡} Observations, &c., at St Helena, edited by Lt. Col. E. Sabine. For 1841–1845, see vol. i. p. 24; for the remaining years the ranges are taken from the tables of hourly observations, vol. ii.

^{page 568 note §} Observations, &c., at the Cape of Good Hope, edited by Lt. Col. E. Sabine, vol. i. p. xvi. From April to September 1841, the observations were two-hourly, afterwards they were hourly. Corrections of +0′·55, +0′·08, +0′·61, and +0′·49, have been applied to the mean ranges, for the months of April, May, June, and August 1841 respectively, on account of the observations wanting at the hour of minimum.

^{page 568 note ║} This epoch is confirmed very nearly by the Milan observations. Schiaparelli's table of the mean diurnal ranges, from 1836 to 1873, given in his memoir, “Il periodo undecennali,” &c., has come to my knowledge only after this paper was written ; they confirm very nearly the epochs obtained from other observations.

^{page 569 note *} Thomson's, Annals of Philosophy, vol. iii. p. 372.Google Scholar

^{page 569 note †} Œuvres de F. Arago, t. iv. p. 427.

^{page 569 note ‡} A similar comparison of the Greenwich two-hourly observations for the four years, 1843 to 1845, gives the range from 8^h 40^m a.m. to 1^h 20^m p.m. = 7′·4, while the mean of the daily ranges =12′·2, whence the ratio—

is exactly that found from the Makerstoun observations. The ratio varies somewhat with the year and the amount of disturbance, the values for the four years at Greenwich being 1·53, 1·76, 1·64, and 1·67. As 1820·5 was upwards of two years from the epoch of maximum disturbance, the ratio found cannot be far from the truth.

^{page 570 note *} The ranges at Greenwich from 8^h 40^m to 1^h 20^m, and at Makerstoun from 8^h 40^m to 1^h 10^m, about the minimum year 1844 were approximately as follows—

We must conclude then, that if the minimum occurred at 1813·7, it had a greater value than in 1844·5 at Greenwich; or, if the value of the former minimum was nearly the same as the latter, that the former occurred probably near 1813·0.

It may also be remarked that the mean of the daily ranges for Greenwich in 1847 was 17′·8, for which year the range from 8^h 40^m a.m. to 1^h 20^m p.m. was approximately 8′·66, which is less than the maximum of 1818. The mean of the daily ranges in 1818 was therefore between 18′ and 19′. From omitting the consideration of the hours to which Beaufoy's ranges refer, it has been supposed that the diurnal oscillations was very small in 1818.

^{page 570 note †} Observations Astronomiques et Physiques faites à l'Observatoire en l'année 1791, p. 345 and note p. 350. Only the observations from 1783 to 1788 were published in Cassini's memoir “Sur la déclinaison et les variations de l'Aiguille aimantée lu à l'Académie Royale des Sc, Août 1871. While the results obtained by Cassini for the mean position of the needle were vitiated by different causes, the deduced ranges are probably approximately true.

^{page 570 note ‡} Different authors have supposed erroneously that Cassini observed only on these four days in each month.

^{page 570 note §} It is difficult in this instance to seek the epoch from the value of the range, as it is not quite certain whether Cassini's ranges are those of the means for each week at the hour of least and greatest declination, or simply the means of the daily ranges. If the former, then the minimum was probably less than that noted, and occurred earlier. It should also be pointed out that we are not acquainted with any marked period, such as that of Cassini, in which the minimum occurs only 2½ years before the maximum.

^{page 571 note *} See the description by Cavendish, , Phil. Trans. 1776, p. 385.Google Scholar

^{page 571 note †} Phil. Trans. 1806, p.416. The ranges for each month were determined from a mean of observations made at those times of the day when the declination was considered greatest and least. Generally 600 observations were made in each month.

^{page 572 note *} “Einige Bemerkungen über die zehnjährige Periode,” &c., Sitz. b. der k. Akad. d. W. 1864. Gilpin's statement is—“Sometimes the needle would be extremely consistent with, itself, so as to return exactly to the same point, however often it might have been drawn aside; at other times it varied 2′ or 3′, sometimes 8′ or 10′, or even more.”—Phil. Trans. 1806, p. 416.

^{page 572 note †} Arago has also indicated the large diurnal oscillation obtained by Gilpin and its varying amount with the season (as elsewhere) as evidence of the free movement of Gilpin's needle. His chief difficulty has reference to the small annual variation of the mean position compared with that found by Casstni, a result which I believe to be wholly in favour of Gilpin's observations, since no such large annual movement as that found by Cassini has been shown by any careful series of observations since his time. See “Œuvres de F. Arago,” t. iv. p. 482. I regret that I have not been able to find the original observations which were made in the Royal Society's Apartments, Somerset House, from 1786 to 1808 by Gilpin, and continued thereafter by Mr Lee, the librarian (See Beaufoy, Annals of Philos., p. 339). It is not improbable, however, that they may yet be discovered.

^{page 573 note *} Comptes Rendus, t. lxxxi. p. 752, 26th Oct. 1875.

^{page 574 note *} Dr Lamont has criticised some of the epochs which Dr Wolf considers certain (“ sicher ”), and has shown that they depend on few observations. He remarks that old observers directed their attention chiefly to large sun-spots; so that Flaugergues (one of the principal observers during the period in question) saw the sun frequently without spots, when many were seen by other observers (“ Einige Bemerkumgen über die zehnjährige Periode,” S. 23). In an interesting investigation on the decennial period of auroræ, sun-spots, and magnetic variations, Professor Loomis has also pointed out the fewness of the observations employed by Dr Wolf, especially those for the years 1802 to 1806 (“American Journal of Science,” April 1873). The numbers for some of these years Dr Wolf has himself marked (?) as doubtful, as in the above table.

^{page 575 note *} “ Astronomische Mittheilungen,” xxxv.; “Vierteljahrsschrift d. Natur. Gesellschaft in Zürich,” 1874, S. 231.

^{page 575 note †} Ast. Mit. xxiv.

^{page 575 note ‡} De Macularum Solis. Ast. Nach. No. 1193, 1859.

^{page 575 note §} Phil Trans., 1870, p. 397. The word is “maximum” at the place cited; but this is evidently a clerical error.

^{page 575 note ║} Trans. Roy. Soc. Edin. vol. xviii. p. 402, 1848.

^{page 576 note *} The principle on which this limitation is founded had been already indicated by me. See “Trans. Roy. Soc. Edin.,” vol. xix. pt. ii. footnote, p. lxxxii. 1850.

^{page 576 note †} “American Journal of Science,” April 1873, pp. 249, 256.

^{page 576 note ‡} I may point out the very few auroræ noted for the maximum of 1818, yet as I have shown (15, footnote) the diurnal range of the magnetic needle for that year was probably as great (or nearly as great) as at any other maximum. It is possible also that the limiting line of equal auroral frequency is variable. In any case little confidence can be placed in the relative magnitude of the number for sun-spots and for aurorae at the epochs under consideration, when compared with those for the half century 1826 to 1876.

^{page 577 note *} See “ Astr. Mitt. xv. Vierteljahrss.” 1863, S. 97, for a notice of a 56 years' period of the auroræ by Professor Fritz; also of Olmsted's period of 65 years. For the period of 80–90 years see “ Ast. Mitt.” xxxviii. S. 378, July 1875. I am not sure that I am acquainted with all the periods which Dr Wolf has discovered, nor do I know if one excludes the other.

^{page 577 note †} It is not possible to reconcile the two results deduced from Dr Wolf's epochs and periods; if we take any of the epochs of maximum given by him before that here employed (1705) we always fail to reach his period of 11 ·11 years ; thus we have

Little weight can be given to these earlier epochs; but whatever weight they may possess, if we start from the sure epoch of 1787 no interval including more than 80 years will be found to satisfy the period of the Zurich, astronomer. It should also be remarked that the longest interval for two successive periods given by Dr Wolf before 1787 is 26 years; since 1818, the longest is 23 years; while from 1787 to 1818 gives 31 years, which, for any other two periods, is an interval unknown in Dr Wolf's sun-spot history. It is obvious that if 10·4 years be near the mean duration, the last result of 11·8 years, obtained on the supposition that there was no maximum in 1797, will go on diminishing, passing through the mean of 11·1 years about 1960.

^{page 578 note *} It will be remarked that this is very nearly the period obtained in 1862 by Dr Lamont (Art. 6), I would therefore repeat that his result was founded on two hypotheses:—1st, That the length of the period should always have been within the limits observed since 1818. 2d, That Gilpin's observations and the sun-spot numbers of Dr Wolf (which did not satisfy the first hypothesis) were worthless. These hypotheses seem inadmissible, and Dr Wolf's result has been in consequence very generally accepted. The whole discussion induces me to believe that a maximum occurred near 1797, and the only point on which any doubt can remain is as to its magnitude,—whether it was really so small as all the observations indicate.

^{page 578 note †} “Analogic de l'Élect. et du Mag.” t. iii. pp. 85, 129.

^{page 579 note *} As the minimum probably occurred, earlier than 1784·8, the ratio here found is too small.

^{page 579 note †} The minimum at Göttingen is estimated from the curve, Plate XXXIX.

^{page 580 note *} This ratio for Munich is that for the true maximum and minimum; the three following ratios are from the greatest and least ranges, corresponding to the middle of each year. They are also not from the same hours.

^{page 580 note †} The greatest yearly mean was the last from the hourly observations, 1848·25 = 11′·69, as the yearly mean range at Dublin increased from that epoch to 1849·0 by 0′·33, and at Munich to 1848·8 by 0·′03 ;.the mean of these two increases has been added to the last yearly mean at Hobarton.

^{page 580 note ‡} The last yearly mean from the hourly observations at Toronto (1848·0) was 11·65 ; the correction to the maximum was derived from the Dublin observations, which showed very nearly the same amount of increase from year to year. Thus, the differences of the yearly mean ranges for the two places, Dublin minus Toronto, were—

The mean excess, 0′·50,has been subtracted from the maximum range at Dublin, 1849·0 = 13′ ·31, to obtain the approximate maximum for Toronto.

^{page 581 note *} This fact is confirmed by the observations made at Lisbon, at 8 A.M. and 2 p.m., by Mr Capello. These give, for the middle of the years, of maximum and minimum—

(“Nature,” April 1876, p. 448.) The first ratio is almost the same as that obtained from the Munich observations for the same hours and years. We may conclude, also, that had the means for the exact epochs of maximum and minimum been obtained, the ratios would have been still greater. On the other hand, the Milan observations do not agree with those at Munich and Trevandrum in showing a nearly constant ratio for different periods ; and the ratios for Milan differ considerably from those for Munich and Lisbon. Thus, the mean ranges for the years 1859·5 and 1866·5, give the following ratios:—

The Milan ratios for the true maxima and minima also vary from

We can explain these great differences of results from two places so near as Munich and Milan, while stations so far apart as Munich and Lisbon agree so well, only on the supposition of some remarkable local cause. We have here another instance of the necessity of having two instruments at the same station to control each other, and to decide what results are instrumental and what not.

The above quantities for Milan are taken from Dr Wolf's Astron. Mitth. xxxviii. p. 382 (July 1875), where I have found, since this paper was written, that the Zurich astronomer has adopted the method of yearly means corresponding to each month.

^{page 582 note *} The generality of this result may also be shown by the equations representing the diurnal variations in different years. These for Hobarton in the minimum year 1843–5 (January to December 1843), and the maximum (observed) year 1848·25 (October 1847 to September 1848) are as follow:—

When we remember the greater amount of irregularity produced in the diurnal variations by the disturbances in 1848, it will be seen that the equations for both years show as nearly the same law as could be expected with a cause whose increased action in producing the diurnal variations is accompanied with increased irregularities.

^{page 582 note †} It is not to be forgotten that near the equator the diurnal variation of the magnetic needle during the equinoctial months is nearly annihilated by the action of two opposing laws, that of the northern and that of the southern hemisphere, one of which prevails more or less in the other seasons. Since both the southern and northern forces vary alike (or nearly so) during the decennial period, it is only the difference of the increments or decrements which are shown at Trevandrum; and it is one of the results most confirmatory of the preceding conclusion that the increase from the minimum to the maximum year bears still the same ratio to the whole diurnal collection at the minimum as for Hobarton, Dublin, and Toronto, where one of the forces always prevails.

^{page 583 note *} Dr Lamont has given the diurnal ranges for different years at several places, deduced from observations at two hours only, which he points out are thus affected by disturbances, and are therefore only “preliminary approximations.” He has concluded from the general agreement of the increase from minimum to maximum that—

where r_n and r′_n are the diurnal ranges for the same (n^th) year at any two stations. Dr Lamont has not, however, given the ratios which result from his data; these scarcely satisfy the equation even approximately. Thus taking St Helena and Munich, for which the ranges are given by him, we find—

I believe, however, that Dr Lamont's conclusion is true, “ that the cause of the ten-yearly period is to be found, in the sun, or more generally, in a cosmic force acting from a great distance.” Indeed, the preceding equation is a general form of that given Art. 36. And when we compare the minimum and maximum at Munich for 1850 and 1860 with those for Trevandrum near the same times, we find—

These ratios are not quite accurate, since only the minimum for 1856·5, and the maximum for 1859·5 at Munich are known to me, but the true minimum and maximum values cannot be very different; these ratios then will satisfy Dr Lamont's equation nearly, and confirm his conclusion. See “ Einige Bemerkungen über die zehnjährige Period, Sitzungsberichte der K. Akad. der Wischenschaften,” München, 1864, II. S. 21.

^{page 584 note *} Astron. Metth. xxxviii. S. 385, July 1875.

^{page 584 note †} This follows from the conclusion, Art. 36, sun-spots appearing only when the magnetic action exceeds a given value.

^{page 585 note *} Trans. Roy. Soc. Edin., vol xxvi. p. 735. See also Trevandrum Magnetical Observations, Art. 398, p. 133, and Appendix, Art. 200, p. 544.

^{page 585 note †} Trev. Mag. Obs., vol. i. p. 120.

^{page 585 note ‡} Derived from Table LI., “ Trevandrum Magnetic Observations,” vol. i. p. 142. The two following means require correction in the table cited. Yearly means for 1864, for 0·406 read 0·414, and mean for July, for 0·399 read 0·406.

^{page 585 note §} It is well known that magnetic disturbances are most felt in high latitudes; it would be of importance to know the ratio of the mean difference (or disturbance) for different parts of the earth's surface; unfortunately this quantity has not been sought in general. Sir E. Sabine has made his valuable investigations on disturbances above certain limits, which vary with the station, so that no exact comparison can be made. The only means from hourly observations, in which all the disturbances are included with which I am acquainted, are those for Makerstoun in the years 1844 and 1845 ; the minimum yearly mean value of the disturbance was that for 1845·4 = l′·66; if we compare this with the minimum at Trevandrum for 1859·7, we have

If we compare, in like manner, the minimum yearly mean range of the monthly mean diurnal variations for the two places, we find

a less ratio than for disturbanees.

Dr Lloyd has given the mean of all the disturbances for the years 1841 to 1850 (Dublin Mag. Obs. vol. ii. p. 88); they are derived, however, from two-hourly observations in 1841–43, and from six three-hourly observations in the following years; and the means are not strictly comparable with those at Trevandrum, we find, however, the ratio of the maximum to the minimum yearly mean disturbance from the three-hourly observations,

which is slightly greater than the ratio for the ranges (Art. 33).

^{page 586 note *} Trev. Mag. Obs. vol. i. p. 142.

^{page 586 note †} There is one remarkably uniform result breaking the regularity of the mean law. The mean disturbance for November in eight years, and for October in the three remaining years is less than that for the months immediately preceding and following.

^{page 592 note *} Proc. Roy. Soc. London, vol. xx. p. 210, March 1872.

^{page 592 note †} It should, be remarked, however, that the discussion of Messrs De La Rue, Stewart, and Lœwy, is founded on more exact and detailed measurements than those for the other periods included in my own investigations.

^{page 593 note *} I may remark that M. Becquerel has proposed a hypothesis by which the atmospheric electricity is derived from the sun, and has connected this derivation with the actions producing solar spots. —Comptes Rendus, Nov. 11, 1872, p. 1126.

^{page 593 note †} M. Faye explains the difference between the velocities shown by the spots in different latitudes and those which should result from difference of radii of the circles of latitude by a hypothesis, according to which the stratum where the falling incandescent rain is converted anew into vapour, has a different form than the external surface of the photosphere,—is, in fact, ellipsoidal, and flattened at the poles. It seems to me that were this double form consistent with dynamical laws, we should have here an indication of a different distribution of solar heat at the equator and at the poles, and a cause of currents between the two, which M. Faye objects to in Sir J. Herschel's hypothesis as non-existent. —Annuaire du Bureau des Longitudes, 1873, p. 516.

^{page 594 note *} Annuaire du Bureau des Longitudes, 1873, p. 527.

^{page 594 note †} Ibid. 1873, p. 525.

^{page 594 note ‡} Annuaire, 1874, p. 460.