THE ORBIT OF URANUS. 



The values of -' ' C D' t R a <lt are formed from ,' D t R by simple multiplication by 

 v, and proper changes of sign. The values of k e and 7c a are then formed by adding 

 the terms of 2 --. f JD' t R dt to the corresponding terms of ; -=-. 



77i *^ '*' C/p 



Perturbations of radius vector. 

 Let us now resume equation (19), and put for brevity 



,, in' (i 



M= Y~r V () 



If we give to u the successive values 0, -4-1, 1, -|-2, 2, -j-3, 3, we have 

 i 2, iV7< (v t v ^ ) 1 & a cos JV+ k, sin 



t c COS| 



k.cos(J\ 



\ k c cos 



f sin( JV ^) } 



-j- 



etc. 



etc. 



etc. 



etc. 



the finite integral being taken with respect to all values of i from oc to -f- oc, 

 and the terms in which the angles Nug vanish being omitted. Proceeding far-, 

 ther to expand with respect to i, if we collect similar terms we shall find the indi- 

 vidual terms in r^p to be as follows: 



> { k c cos N-\- k a sin N \ 



+ i 



etc. 



etc. 



-f (/>!?>+ Mi) ("i 



+ (M4-M) ( v 

 -f- etc. etc. 



| k, cos (JV ff ) + k, sin (^ 



+ ^ 



/> cos 



k, sin (N -J- 2g) 



etc. 



etc. 



{ 



M ^ + (Ms +B?0 fa 

 etc. etc 



-(- 



etc. 



(v 2 

 etc. 



| k c cos (N 2<7) -f k, sin ( JV 2gr) \ 



cos (N+ Zff] + k. sin (N + 8*7) f 



