VOL. L.] PHILOSOPHICAL TRANSACTIONS. 173 



, . .V sin. A E X EAr , ^ sin- ae ^ ^ ak X ok 

 (:: sin. EAe : sin. ec) :: EAe : Ee = : =i A X -.- — X :, — = 



^ ^ SUl. E. Sin. F OA^ 



k X ^"^' ^^ ^ ^ -°---- — ^~ — . But in the triangle eh a, right-angled at a (where 



HA is supposed to represent the sun's declination, ae his right ascension, and 

 HE his distance from the equinoctial point e*) we have (per spherics) 



sin. A e: 1 (rad.) : : co-t. e: co-t. ah, 



(sin. ah)*: (sin. eh)^:: (sin. e)'^: l^(rad)^ 

 Whence we get, sin. ae x co-t. ah x (sin. AH)"^=(sin. eh)^ X co-t. e X (sin. e)*. 

 But co-t. AH X sin. AH = co-s. AH X 1 (rad.), and co-t. e X sin. e = co-s. e X 

 1 (rad.): therefore sin. ae X co-s. ah X sin. ah = (sin. eh)* X co-s. e X sin. 



J .1 7 V sin, AE X co-s. AH X sin. AH , ^^ v> / • \'2 



e; and, consequently, k X ■■ =« X co-s. e X (sm. eh)^= 



Ee. 



Let now the sun's longitude eh be denoted by z, considered as a flowing 

 quantity; then, (sin. z)'^ being = -^ — -^ co-s. 2 z, we shall have k X co-s. e X (sin. 

 eh)*= -'q: k X COS. e X (1 -^co-s. 2 z). But the angle described about the axe of 

 rotation vp, in the time that the sun's longitude is augmented by the particle z, 



will be = — X Z. Therefore, by the general proposition, as 1:1. k x co-s. 



X E (1 — co-s. 2 z) : : y X Z : 4- ^ X Y X co-s. e X Z — Z co-s. 2 z, the true 

 regress of regress of the equinoctial point e, during that time: whose fluent 

 4- ^ X — X co-s. ex (z — 4- sin. 2 z), will consequently be the total regress of 

 the point e, in the time that the sun, by his apparent motion, describes the 

 arch he or z ; which, on the sun's arrival at the solstice, becomes barely = ^ ^ x 



— X co-s. E X an arch of 90° , the quadruple of which, or -i- ^ x — X co-s. e X 

 360° (--^X — ^~r^ — X co-s. e X 360^) is therefore the whole annual pre- 

 cession of the equinox caused by the sun. This, in numbers (taking ~.= SS2\ 

 comes out ~-~^,. x J57 X O.917176 X 36o° = 21'' 6". 



M. Silvabelle, in his essay oh this subject, inserted in vol. 48 of these Transac- 

 tions,-}- makes the quantity of the annual precession of the equinox, caused by 

 the sun, to be only the half of what is here determined. But he appears to have 



* No error arises from considering the triangles EAe and aeh, as being formed on the surface of 

 a sphere, though the earth itself is not accurately such. The angle eao representing the effect of 

 the solar force, is properly referred to the surface of a sphere ; therefore after its measure is truly 

 determined, the figure a pap is itself taken as a sphere, in order to avoid the trouble of introducing a 

 new scheme. — Orig. 



+ Page 436, Vol. 10, of these Abridgements. 



