THE ORBIT OF URANUS. 23 



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

 order terms by means of this equation will depend upon circumstances. 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 values of all the quantities whicli 

 enter into the equations (12), (13), (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 first order, and afterward add those of the second. There is, how- 

 ever, 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 (^p and tip'. Tliis is eff'ccted 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. This is in fact what we have supposed to be done in the 

 preceding investigation. The values of ^v, liV, hp, tip' in formula (11) will then 

 contain only periodic terms. 



In computing the terms of the first order we determine the value of ^p from the 

 equations (19) and (20), using the value of Q(, in (lb). Then those of tv are 

 obtained by integrating the equation 



-,,=,-, -J -„ (/< — 2mcos4'-2- y~^) 



at 1 -|- m*' ^v Tj 



Having found the values of ^v and hp for both planets, tliey are to be substituted 



17? '^ 7? 



in (11), to obtain hB, h and S -. But, rigorously, ^v and ^v' axe not the 



dv dp 



same with Sv and hv\ owing to the movement of the orbits of the planets, and the 

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

 perturbations of tlie second order, which depend on tv, hv\ hp, and hp, we may 

 use the following equation for (•R, and similar ones for its derivatives: 



dR , SB , , dB . , dB . , ■ ,^.. 



^E = ^^- ^v + ^ ^v'+ ^~ k + ^k' (20) 



dB 

 Having thus found hB, and hence D\^B by difi"crentiation, and then h - — , we form 



dp 

 (26) 



the quantity 



which is the difl"erence between the value of Q^y in (IS) and tlmt of Q in (12). 

 The terms in hp arising from hQ are then to be computed by the formula; (19), 

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

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

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

 of the former is really ^p — f?p, we find, neglecting qTiantitics of the third order, 



^ 2 dh'v 



an' \fh -|^ dt - 2hpS^f^ dt | - 2n cos ^ (^^p - hp') 



