Till: OKI! IT OF URANUS. 23 



The most convenient mode of making the numerical computation of the second 

 tinhr terms by means of this equation will depend upon ei re n instances. If the 

 perturbations of longitude and radius vector of both planets are already known with 

 a sufficient degree of approximation for the computation of formula (11), it will be 

 more convenient to form at once the complete \ allies of all the quantities which 

 enter into the equations (lv>), (l:J), (19) to 22), and (23), so that no steps of the 

 process shall have to be repeated. If such perturbations are not known, they 

 must first be computed, and it will then be necessary to begin with the perturba- 

 tions of the lirst order, and afterward add those of the second. There is, how- 

 e\ t r, one class of terms of the second order which it will be most convenient to 

 take account of from the beginning, namely, those arising from the constant term 

 in A() and y. This is (dec ted by correcting the mean distances for an approximate 

 value of these constants at the beginning of the computation, and then proceed- 

 ing in the usual way. 'I'll is is in fact what we have supposed to be done in the 

 preceding investigation. The values of iv, iv 7 , ip, ip' in formula (11) will then 

 contain only periodic terms. 



In computing the terms of the first order we determine the value of ip from the 

 equations (19) and ('20), using the value of Q 9 in (18). Then those of v are 

 obtained by integrating the equation 



d&v awVf 1 CdRo,. . fy 



= l + mJ 5F*"* 1 "**?' 



1 1 \ ing found the values of iw and ip for both planets, they are to be substituted 



J5 15 *!) ^? 



in (11), to obtain &R, i - and i - -. But, rigorously, iv and iv 7 arc not the 



<9v dp 



same \\ ith Aw and it/, owing to the movement of the orbits of the planets, and the 

 corrections for iy are also to be added. Considering, for the present, only the 

 perturbations of the second order, which depend on it), it/, ip, and ip', we may 

 n M- the following equation for i.R, and similar ones for its derivatives: 



f)Tt 



Havinir thus found i.R, and hence U^R by differentiation, and then i , we form 



op 

 the quantity 



which is the difference between the value of @ in (18) and that of Q in (12). 

 The terms in Sp arising from i@ are then to be computed by the formulae (19), 

 (20), (21), and (22), when we shall have ip accurate to quantities of the second 

 order. Let us represent these additional terms by d*p. Subtracting (24) multi- 

 plied by r, 1 from (23), recollecting that the ip which appears in the second term 

 of the former is really ip i*p, we find, neglecting quantities of the third order, 



= ' { f i ** dt - Mp/'. ff " dt } - 2n cos * (i'p - 

 at 0v dv ) 



