22 THE ORCIT OF URANUS. 



to which expression is to be added, in lieu of tlie terms wliich will have infinite 

 values of v as a factor. 



1 m'ar^-irt" 



4 a^{\-\-m) 



U^pJc^'c'-^^qJ^'n (2-3) 



771 'ill 



¥"^ and A;*','' being the factors of — nt cos %((j and at sin ?/r/ in the expression 

 for Q. "' "-' 



The formulae 19, 20, 21, and 22 give the complete expressions for the perturba- 

 tions of the logarithm of radius vector by successively substituting in it all the 

 terms of Q. 



Perturbations of Longitude. 



We now pass to the perturbations of longitude. In the Mecanique Celeste 

 (Premiere Partie, Liv. ii. Chap, vi.), Laplace gives an equation (Y) by which the 

 perturbations of longitude, which are of the first order, may be derived from those 

 of the radius vector without the formation of any other derivatives of R than those 

 which enter into Q. But the formula does not seem easily adapted to the case in 

 which the perturbations of the second order are taken into account, we shall 

 therefore derive all the perturbations of longitude from the second of equations (1). 

 By integration this equation gives 



dv u ( rSR , ) 



dl = ?V-dV''+''\ 



C being the arbitrary constant of the integral. Representing, as before, by sub- 

 script zeros tlie values of the co-ordinates corresponding to the ellipse to which the 

 orbit is supposed to reduce itself when the disturbing forces vanish, we have 



dvQ a?n cos -^ ^C 



dt ~ 9V "" '"o^ ' 

 because the constant to which the integral must reduce itself in the elliptic motion 



is — —. Subtracting the last equation from the preceding, and putting 



V — v^^ hv, we find 



dhv 



5» """ ^^>i.-i..i.5 





Developing ^.- to terms of the second order Avith respect to the disturbing force 



^ = ,^ (1 - 2^V + W - "^tc-), 

 which, being substituted in the last equation by putting 



= a'r. 



gives 



"-^'^ = IT^ (1 - 2^P) /^/'' - 2" ^«« ^ (^P - V)' (23) 



which is rigorous to quantities of the second order. 



