^{page 511 note *} Trans. Royal Soc. Ed., vol. xvi. p. 74. See also “Introduction to Makerstoun Observations.”

^{page 511 note †} See Trans. Royal Soc. Ed., vol. xxii. p. 467.

^{page 513 note *} If we correct the yearly means for a regular secular change, the remaining quantities will show a maximum and minimum nearest the epochs of minimum and maximum disturbance, and with an interval of about five years.

^{page 516 note *} Copy of Abstracts of Observations made at Simla (lat. 31° 6‘ N., long. 5h. 9m. E. of Greenwich) under the direction of Colonel Boileau, is in the Library of the Trevandrum Observatory. I have not been able to discuss these observations satisfactorily, as the temperature of the magnet is not given, excepting for 1842. I have, however, taken some pains to arrive at an approximate correction, and can state as results that at Simla the horizontal force increased throughout the whole period. The curve for 1842 is projected, Plate XXIII.; but the range of temperature in the bifilar box is too considerable to allow us to put much value in the conclusion that may he drawn that the variations of the monthly means are less than at other places.

^{page 517 note *} 1845, Trans. Brit. Assoc.

^{page 517 note †} 1846, Jan. 5, Trans. Royal Soc. Edin., vol. xiv. p. 99. The Toronto and Makerstoun results are projected in Plate III., vol. xiv.

^{page 517 note ‡} Makerstoun Observations, 1844, p, 357 (foot note).

^{page 519 note *} See Bifilar Magnetometer, Trans. Royal Soc. Ed. vol. xxii. p. 467.

^{page 519 note †} Resultate des Mag. Obser. in München, 1843–4–5. See also Makerstoun Observations, 1844, p. 357 (foot-note); and General Results (Trans. Royal Soc. Edin., Vol. xix., part ii.) p. xxxiii. For convenience of comparison, I shall repeat here the quantities projected in Plate XXIV, and to be found in the works cited:—

^{page 520 note *} It should be noted that the monthly means depend upon hourly observations, excepting in the following cases, which depend on two-hourly observations:—Toronto, 1841 till July 1842; St Helena, 1841 till September 1842; Trevandrum, 1841 till February 1842; Singapore, till June 1842; and in the case of Makerstoun, where the monthly means 1841–2 depend on 4 three-hourly observations; 1843, on 9 two-hourly observations; 1846, on 9 two-hourly observations; 1847, on 5 three-hourly observations; and 1848–9, on two observations daily. For this reason the means at the different places are not strictly comparable.

^{page 520 note †} I have corrected the Toronto Observations for 1846, 1847, and till June 1848, at which date the observations stop in vol. iii. (received since this paper was written). These are added in Table VII., and have been projected (Plate XXIII); they show so considerable a resemblance to the observations at Hobarton and Makerstoun as to render it probable that the different character of the movement in 1844–5–6 was due to instrumental causes, a supposition which is rendered more probable by the great difference between the temperature coefficient derived from the usual observations and from hot and cold water experiments.

^{page 522 note *} Makerstoun Observations 1844, p. 395 (foot-note).

^{page 524 note *} It may be supposed that the increase everywhere shown in the end of 1841 might have been in some way connected with the commencement of the series; but I think there is no ground for this supposition. The observations were commenced early in 1841; the magnets were removed monthly from their stirrups for observations of absolute intensity; and any tendency to stretch in the wires would have been got rid of in this frequent manipulation. At Toronto, indeed, the second month projected (September 1841) shows a diminution of force. I think there is every probability that the increase was a real increase of horizontal force at all the stations.

^{page 525 note *} I adopt this in preference to giving the quantities with the algebraic signs + and −, since some persons in glancing at such results may fail to perceive a maximum if it has the negative sign prefixed.

^{page 529 note *} The usual method to determine the epochs of maxima and minima is to obtain by approximation the roots of the equation to the maximum or minimum. The following method has been adopted by me for the computation of the above epochs:—

Let the general term of the equation (1) to the curve be

p_i = a_i sin (iθ + c_i),

and let

q_i = a_i cos (iθ + c_i).

If, then, we obtain an approximate value of θ for the maximum (or minimum) from the projected results (observed or calculated), and substitute this value in the above equations, and if α be the correction of the approximate value of θ to the true value, we shall have

or, for a very accurate determination,

In the same way we may obtain the correction α′ of the approximate epoch for the mean value of y from the equation

and for points of contrary flexure, which ought not to be neglected in some cases, we have the correction α″ to the approximate epochs substituted as before in the values of p_i, q_i.

These values are easily calculated with the aid of a table of natural sines to three decimal places for every 15′ of the circumference, and of a table of multipliers like Crelle's.

^{page 530 note *} The means of the Toronto Observations, July 1846 to June 1848, corrected for secular change at the rate of −1·33 per mensem, are projected in Plate XXIV.

^{page 533 note *} I have taken the trouble to deduce the absolute force for the year 1845, the only year for which the data are in my possession, so as to eliminate the error due to the bifilar temperature coefficient. The following are the values in absolute measure from the Toronto volume, and according to my corrected computation:—

The law is evidently to be traced even in these results; the greatest error of a monthly mean is that for July, = 0·0038, or in ten-thousandths of X, = 10·8. The errors are chiefly due to the difference between the mean temperature of the bifilar for the three days on which the absolute observations are made and for the month; in July, for example, this difference is 9°·0 Fahr.

^{page 534 note *} See Cape of Good Hope Magnetical Observations, vol. i. Table xxxvi. p. lxx.

^{page 535 note *} Cape Observations, vol. i., p. lxxi.

^{page 535 note †} The bifilar temperature is not given with the observations of absolute intensity. In attempting to determine this temperature by the date and the ordinary hourly observations, it appears to me that there are several errors in the tables. Thus, as to the dates 1847, March 2^d 0^h, August l^d 0^h, September 5^d 0^h, October 3^d 0^h, &c. &c, of Hobarton time, they are all noon of Sunday, when no observations were made; the days are perhaps Göttingen time, and the hours Hobarton time? Again, the bifilar readings given as those during the observations 1846, April 3^d l^h; 1847, February 2^d 0^h, March l^d 0^h, April l^d 0^h, December 3^d 0^h, December 4^d 0^h, &c, appear to be erroneous; in some cases, perhaps, the error is one of 10 divisions.

^{page 536 note *} This is the same as if each whole observation had a value, since each complete observation was made up of two deflections, excepting in 1846, and January to March 1847, when three deflections were made to complete the observation.

^{page 538 note *} The value of the unit coefficient of the bifilar magnetometers at Makerstoun and Trevandrum was very nearly k = 0·000140. As the results in this paper were first deduced from the observations at these two places, the same value of the unit was adopted for the other places in these tables.

^{page 554 note *} Lat. 55° 34‘ 45“ N., Long. 0^h 10^m 3·5^s W. Height above the sea, 213 feet. Observations Trans. Royal Soc. Edin., vols. xvii., xviii., xix.

^{page 555 note *} In this Appendix the units are always ten-thousandths of the force, unless when sc div. (scale divisions) is affixed.

^{page 557 note *} The Cape means not employed, as the readings were descending rapidly by secular or instrumental change at the time.

^{page 557 note †} Lat. 33° 56‘ S., Long. l^h 13^m 56^s E. of Greenwich. Observations, vol. i, printed under the superintendence of Lieutenant-Colonel E. Sabine.

^{page 558 note *} Lat. 15° 56‘ 41“ S., Long. 0^h 22^m 42^s W. Height above sea, 1760 feet. Observations, vol. i., printed under the superintendence of Lieutenant-Colonel E. Sabine.

^{page 558 note †} Adjustment, &c., St Helena, vol. i. p. 33.

^{page 559 note *} Lat. 42° 52‘·5 S., Long. 9^h 49^m 50° E. Height above the sea, 105 feet. Hobarton Observations, vols. i., ii., and iii., printed under the superintendence of Lieutenant-Colonel E. Sabine. The temperature coefficient employed for 1842–43 was q‘ = l·50 sc. div., instead of the more accurate value 1·52 (see Trans. Royal Soc. Ed., vol. xxii. p. 481).

^{page 559 note †} Hobarton Observations, vol. i,, Adjustments, &c, p. xliii.

^{page 560 note *} The error here noted is much more marked in the case of the hourly means for each month, where the same method of summation and division is followed. With some classes of observations (as those of the barometer and thermometer), the hourly means are scarcely comparable through these omissions.

^{page 560 note †} Lat. 8° 30‘ 32“ N., Long 5^h 7^m 59^s E. Height above the sea, 200 feet. Observatory of his Highness the Rajah of Travancore; John Caldecott, F.R.S., Director 1833—1849. The observations in manuscript are in the archives of the Royal Society of London.

^{page 560 note ‡} The agreement between the Trevandrum and Singapore means is so considerable, that for all ordinary purposes, a few days wanting in the one may always be supplied from the other without any risk of error.

^{page 562 note *} Lat. 1° 18‘ 32“ N., Long. 6^h 55^m 46^s E. Height above the sea, a few feet. Observatory of the East India Company; Director, Captain C. M. Elliot, F.R.S. The observations in manuscript are, it is believed, in the archives of the Royal Society. Copies of some of the observations are also in the Library of the Trevandrum Observatory. The observations have also been printed, but not published as yet.

^{page 562 note †} It is not improbable that there is an error in the mean for May 20 of 10 sc. div. (k = ·0001946), or 13·9 sc. div. of the table.

^{page 564 note *} Errors from this cause were found to have occurred at Trevandrum.

^{page 565 note *} Lat. 43° 39‘ 45“ N., Long. 5^h 17^m 26^s W. Observations, vols. i. and ii., published under the superintendence of Lieut.-Colonel E. Sabine.