VOL. LXIX.] PHILOSOPHICAL TRANSACTIONS. 581 



tion; and therefore the said point of the equator, at the end of that time, will 

 have acquired a velocity which would carry it through ^^"^ in the same time. 



^ 7. Let T represent the time of the earth's revolution in its orbit, / the time 

 of its rotation round its axis, and suppose » to be a small arc similar to i in a 

 circle whose radius is unity. In fig. 5, let aq be the equator, and take aL 

 equal to y, and bt, perpendicular to A.b, equal to 3pmn-w', and aZ;, bl, will repre- 

 sent the directions and quantities of the two difterent motions of the point a, 

 and consequently At will be the direction of the new equator, and as a^ or At is 

 to bt, so is the radius unity to the sine of the angle tAb; and if aq or ag be taken 

 equal to a quadrant, gq, the measure of the angle gaq, is equal to ^'"""^ . 



§ 8. Suppose s the sun's place in the ecliptic ns, n the equinoctial point, na 

 the sun's right ascension, and ro a perpendicular on an; then ru is to gq as the 

 sine of an to the radius, and 7N to ro as the radius to the sine of the angle at n, 

 the inclination of the ecliptic to the equator, and, ex aequo perturbate, rs to gq 

 as the sine of an to the sine of n, and rN the small precession of the equinoxes is 



equal to 3pmn X - - X — . . 



^ ' T sin. N 



^ 9. In the spherical triangle asn, the sine of an = cotang. n X tang, as = 



m X COS. N J . , . /. m , . , 3pt X (sin. a;) ' x «■ x cos. n 



: , and the sme of sn = - — , also rN is equal to ^^ ■ :— , 



n X sin. N sin. n ^ t ' 



whosefluent,or 7^ X {Iw — sm. 2w) X cos. n, gives the precession of the equi- 

 noxes during the sun's motion through the arc ns of the ecliptic: when ns is 

 equal to a circle, then the whole fluent becomes equal to — — x cos. n, and as 



2t is to 3pt X COS. N, so is 60 X 60 X 36o to 21" + 6'" the annual precession 

 of the equinoxes in seconds produced by the sun's attraction. 



^ 10. We might now proceed in a similar way to investigate the effects of the 

 moon's disturbing force, the rotation of the earth's axis, and the equation of the 

 precession; but since these propositions are purely mathematical, and the com- 

 putations have already been gone through by other authors, it will be needless to 

 repeat them here. 



^11. Newton was the first who attempted to explain the precession of the 

 equinoxes from its causes. Since his time various other solutions have been 

 given by the most celebrated mathematicians ; and it deserves to be noticed, that, 

 in a case where there can be little doubt that he was mistaken, other authors 

 have found it difficult to agree among themselves in differing from him. Mr. 

 D'Alembert, in the year 1749, printed a treatise expressly on the subject, and 

 has since said,* that himself is acknowledged to be the first who determined 



* D'ailleurs, des georaetres, vraiment capables d'apprecier mon travail, ont abondamment supplee 

 a tout autre temoignage, en declarant que j'ai ouvert le premier la route pour resoudre ce genre de 

 equstions. See Opusc. Math. vol. 5, sur la Precession des Equinoxes.— Orig, 



