THE ORBIT OP NEPTUNE. 37 



Peirce and Kowalski, as may be seen by reference to the preceding values of their 

 coefficients. They are, in fact, very nearly confounded with the elliptic motion 

 of the planet, but not exactly. We shall, at present, retain only the small resi- 

 duals, after subducting those portions which are sensibly elliptic. The entire 

 terms are as follows : 



1. In the longitude. 



Action of Uranus, + 0".385 sin Z — 0".092cosZ — 0".014sin2Z — 0".002cos2Z 

 Saturn, + 0.099 sin Z — 1.412 cos Z — 0.018 sin 2 Z — 0.020 cos 2 Z 

 Jupiter, +2.393 sinZ — 0.567 cos Z + 0.018 sin 2 Z — 0.029 cos 2 Z 



Total, +2.877sinZ — 2.071 cosZ — 0.014sin2Z — 0.^51 cos2Z (a) 



2. In the logarithm of radius vector. 



Action of Uranus, + 1 sin Z +14 cos I 

 Saturn, — 34 sin Z 

 Jupiter, — 11 sin Z — 51 cos I 



Total, —44 sin Z —37 cos Z (l) 



■ Changes in the functions e sin it and e cos n, represented by hli and hh, will pro- 

 duce the following changes in the longitude and log r, 



hv =2 Sk sin Z — 2 ^Ji cos Z + | {Uk — UJi) sin 2 Z — | {kU + Uh) cos 2 Z 

 h log r= — MU sin I — Mhh cos L 



Taking the elliptic terms to be subducted so that the coefiicients of sin Z and cos I 

 shall vanish, we must put 



M = + l".036; 57^ =r + 1".438, 



which will produce the inequalities 



hv —+ 2".877 sin I — 2".071 cos Z0O".OO7 sin 2 Z — 0".037 cos 2 Z 

 h log rz:z — 21 sin Z — 30 cos Z. 



Subtracting these elliptic inequalities from (a) and (5), we have for the residuals 



hv — — 0".021 sin 2 I — 0".014 cos 2 Z 

 h log r = — 23 sin Z — 7 cos I. 



So that the constants of P^, etc. are 



Constant of Psi = 



Pol = 



P^^ - _ 0".021 

 P,2 = — .014 

 i?,i = — 23 



