

THE ORBIT OF URANUS. 



In the terms of SR introduced by the perturbations of Saturn, namely 



•0 7? *) /? 



- , fe' 4- „ , ^p', tlic differentiation represented by D\ should be performed bv con- 

 ov op ' •' 



sidering hv' and Sp as constant, altliough they are expressed as a function of the 



mean longitude of Uranus, as well as of Saturn. The mean longitude of Uranus 



thus introduced is therefore represented by U', which is regarded as constant iu' 



taking -D',/?, and ZJonly supposed to vary. 



dli 



6R 



Again, in the terms — — ^v -|- - ^p, since Iv and ^p represent perturbations of 



Uranus, their complete derivatives, with respect to the time, are to be taken. But 

 their expressions contain the mean longitude of Saturn as well as Uranus. The 

 mean longitude of Saturn thus introduced is represented by *S", and is to be con- 

 sidered variable in obtaining D\hB, while S is considered constant. The ratio of 

 the coefficient of t to n in the various terms of this part of hR is given to the 

 right of each corresponding term. 



The value of DfiR being once obtained, there is no longer any distinction 

 necessary between f^, U\ or between S and *S". The similar terms are therefore 

 combined by putting *S" = S; U' = U. 



6R 

 <9p ' 



^hQ^^'^jhD\Rdt+'^ h^ 

 m m ^ m dp 



and of the other quantities Avhich enter the perturbations of the co-ordinates. "We 

 sliall begin with those terms which depend only on the mutual action of Saturn 

 and Uranus, because they are few and small, and the only terms which are sensible 

 are those in which the coefficient of the mean longitude of Saturn is — 1. We 

 shall therefore confine ourselves to these. And, instead of employing the con- 

 densed formula^, we shall make the computation in full by (13). 



From the above values of 2bD'i R and 2^ — ^'"', we form She following value of 



