76 



THE ORBIT OF URANUS. 



In the terms of 55 introduced by the perturbations of Saturn, namely, 



JO 75 JO J2 



5' + -.- 5p' the differentiation represented by D' t should be performed by con- 



dV c?p 



sidering 5v' and 5p' as constant, although 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 in 

 taking D t R, and C/"only supposed to vary. 



Again, in the terms 



f)R 

 Jv -\ 5p, since v and 5p 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 $', and is to be con- 

 sidered variable in obtaining D' t &R, while S is considered constant. The ratio of 

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

 right of each corresponding term. 



The value of D' t &R being once obtained, there is no longer any distinction 

 necessary between U, IT, or between S and *S". The similar terms are therefore 

 combined by putting S' = S; IT = U. 



^ y-> 



From the above values of 25Z>' ( R and 25 - 



we form the following value of 



2", 



m " m <9p 



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

 shall 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 formulae, we shall make the computation in full by (13). 



