30 THE ORBIT OF URANUS. 



quantities of the first order, in which case and r may be assumed equal. These 

 values are 



8<p = sin -T 8p 4~ cos r lq 

 sin <> 8r = cos q> (cos t 8p sin T 8q) 



the terms dependent on 8p' and 8q' being omitted because, being purely secular, 

 they may be included in the mean values of fy and r. Substituting in the expres- 

 sion for 83 



cos 383 = cos <> | sin v 8q cos v Ip \ . (37) 



In the case of all the larger planets both cos 3 and cos $ may here be put equal 

 to unity, when the expression for 83 will become 



83 = sin v lq cos v p. (38) 



To develop this expression in purely periodic terms we must substitute for v its 

 value in terms of the mean longitude or mean anomaly, namely, 



v = I -\- 2e sin g -(- e 2 sin %g -\- etc. ; 



suppose the terms of 8p and 8q depending on any argument, N to be 



8p a s sin jV a t . cos N 



8q = ', sin N -\- a' c cos N 

 and put 7i for the longitude of the perihelion, so that 



then, to terms of the first order with respect to the eccentricities, we have 

 83 = e (a e cos 7t -\- ' sin 7t) sin N e (a c cos it + ' c sin 71) cos N 



_i_ 1 $ (a a _|_ a '^ cos n -\- (a' s a e ) sin TI\ sin (^V-f- g) 



~\~ I I ( tt a ') cos ^ 4~ ( ft ' 4" a *^ s ^ n n \ cos (-^"4" S') 

 4-| \ (a, a' c .) cos 7t 4- (' 4- a o) sin 7t| sin (.V - #) 

 _i_ i j ( a<! _^_ d^) C os n 4~ (' ) sin 7i I cos (N- g) 



4~ \ e \ (a, 4- a' c ) cos TI 4~ (' s i n 7t j sin f Ar ' ^^ 

 -\- % e \( a c a ') cos " 4~ ( a 'c 4~ a ) s ' n ^ I cos 

 4~ 2 ^ S (ct, ot',,) cos 7t 4~ (^ 4~ "'c) sin t j sin 

 -|- 1 e{ (a, -j- a',) cos n 4- (a' c a,) sin n \ cos 



The point of the orbit from which n and v are counted is entirely arbitrary, 

 and, in considering the action of but a single planet, it will be most convenient to 

 count them from the common node, in which case n must be replaced by u, and 

 8p and 8q by 8k and 8vj. Thus, deducing the perturbations of the latitude imme- 

 diately from the formulae (27), we shall have 



&3 = sin v 8rj cos v 8k. 



