THE ORBIT OF URANUS. 



21 



and use the symbol ;■„ as before, to represent the ratio of tlie mean motion of the 

 planet to the coefficient of t in the angle N-\-i<j, so that if ^n' represents the 

 coefficients of t in iYwe have 



n 



11 

 II -\- in 



n 



+ i 



wc find the expression 



fc Qdt= \ -^^'^ I {I'f /.■, - rJ.-M) cos (.V+ !^) + (vl, h - ra\nf) cos (N - ig) 

 (rf/.-, + 2'/.-,«0 sin (.y + ig) + (j^i, Z-, + r_,hn() sin (iV^ — /^) I 



fnQdt = J i^^'^«'- I (r//,-, - 7.A.nO sin (X+Jg) - (vljl; - vj.-jif) sin (N-jff) 



- O'/h + r,hnf) cos (.V+^V/) + (^^A + »'_/'V»0 cos (.V-y^) J . 

 If we now put for brevity 



vrj,; — v,l\nt = Si, 

 the general value of r^h^ becomes 



r (0 - ^. ) cos (.Y+ (*• +/) i/) + (.. - .9, ) sin (.Y+ (; +y) g) 

 , J + (c. - ^^-.O cos (.V+ {i -J) g) + (.9_, - s, ) sin (^+ (^ -/) ^) 

 ^*' I + (''. - <^-0 cos (.V- (/ -j) g) + (.._, - sj ) sin {N- (i—j) g) 



L + (c_,- c_,) cos (.Y- (i +y) i/) + (,9_, - «_,) sin (Y^- (/ + y ) g) 



If, as before, avc transform this expression by putting 



in the first line j = u — i ; 



the value of Vj-^p reduces to 

 1 m'a 



V 2 I Pi9(u-t) (c,«-, — c, ) cos (Y^+ 7/^) +iJ,<7,„_o («, — .^._o) sin (.Y+ ?/r/) ) 

 or, putting i — n for t in the last line, 



24.- 1 i>.5.«) -Piu-i,g> \ 1 (c,,-,)- c,) cos ( A^+ ?///) + (.9, - .9„_,) sin (7^^+ wr^) \ ; 



(21) 



