Til I! OH HIT OF URANUS. 



21 



ami MM- the symbol > as before, to represent the ratio of the mean motion of the 

 planet to the coefficient of t in tin- angle N-\-ig, so that if /u' represents the 

 coefficients of / in \ \\e Iia\e 



V = 



n 



\M find the expression 



' '''" "',,~,r { (''* / '-.- l ''*V<Oc 



(i-, 1 *. + vfcnt) sin (N-}- ig) + (vl* It. + r_,*.0 n (JV- t 



r//,- c - ,./-./) sin (JV -\-jg) (v^c. v^.nt) sin (N- m 



v * 1 ''. + 'A') C08(tf-f y^) -f (vljc. + v_/.- c nO cos(N-j'g) 1 . 



If we now put for brevity 



r?l; -\- vjtjit = c it 



v?k, rjf.nl = 8 H 

 the ireneral value of r, 2 (Sp becomes 



. 1 n'arr* x 



(c. -c t ) cos (.V+(;+/) */) + (/., -, )sin(^+(+y) i/ )] 



(c, O cos (N-(i -j)g) + (_,-> )sin(iV-(i y)7) 



.< c_j) cos (iV (i -\-j)g)-\- (s-j _<) sin (N (i-\-j 

 If, as before, we transform this expression by putting 



in the first line j = u i ; 

 in the second " j = * ; 

 in the third " / = *+; 

 in the fourth " j = i ' 



the value of r, l fy reduces to 



1 Wl'rt 



X 



- cos 



g 



or, putting t for i in the last lino, 



_ i 

 = 



- c ') cos 



' -') 8in 



