346 Prof. Bessel's Additions to the Theory of Eclipses, 



A«? A &> &c. are here assumed as expressed in parts of the 

 radius ; they must, therefore, be divided by co = 206265 if they 

 are meant to be given in seconds. The coefficients a, b ...a! b' 

 are the differential quotients of P — u, and Q — v in relation 

 to «, 8, 7r and e* ; neglecting in their values the small quantities 

 of the order of a — A and 8— D, which, on account of the small- 

 ness of A «> A 8, &c. will produce no error of any conse- 

 quence, the expressions for them will become very simple: 



COS S , „- : , ■ ,. 1 P I 



a = - — ; a! = 0; b = ; b' — -. — -; c = — - — — ; c = 



sin vr sin ?r tang tr 



; <Z = — — - ; d! = -^- . Adopting these expres- 



tang ?r ' d. e» ' d.e a t & r 



sions, and substituting w sin N and 7? cos N for p' and q\ and 

 # for : — , where s represents the number of seconds 



a . n . sin it £ 



equal to the hour to which p' and q' belong [6], we obtain 

 i as 7* sin N . cos 5 A « + A cos N A 8 — A cos tt . A ?r [P sin N 



+ Q cos N] - h co sin «■ . A 4"-^ sin N+ -^- cos N 1 



J L d . e* d.e i -J 



i'-ss — ^ cos N . cos 8 . A « + 7* sin N . A 8 + h cos x . A t 

 [P cos N - Q sin N] + h.to sin tt . A ■ e* [-^- cos N - 

 —^- sinN~|. Hence, 



( / + la^) i T ± = - cos ( N + ^- cos5 -A a +sin(N + ^)A8 

 + costt. Att[Pcos(N + 4/) — Qsin (N + \p)] + w sin tt A? 9 X 



[^ cos( N+^)-^sin(N + ^)]. 



The part dependent on At may be reduced to a more con- 

 venient form, and the one dependent on A- e* may be further 

 developed. Substituting in the former p + n sin N.Ti and 

 q + n cos N.T'for P and Q, it becomes costt.At[pcos(N + \J/) 



— q sin (N + 40 — ?i . T' sin 40 = cos tt . A «r [ (p cos N 



— q sin N) cos ^ — (p sin N + £ cos N + w T') sin ^] ; and 

 further, making x = y sin N — p cos N, in = « T — p sin N 



— §- cos N, and putting for T' its expression *~ d ~ T > it as- 

 sumes this form 



— cos tt . A tt I~x cos 4> + ^-(t — d — r) sin 4/1 



It will be easily perceived that x. to sin ?r is the smallest di- 

 stance of the true path of the moon from the star, positive if 

 the moon passes to the north, negative if she passes to the 



south 



