THE ORBIT OF 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 sinZ 0".092 cosZ 0".014 sin 2 1 0".002 cos 2 1 

 Saturn, + 0.099sinZ 1.412cosZ 0.018sin2Z 0.020cos2Z 

 Jupiter, +2.393 sinl 0.567 cosZ + 0.018 sin 2? 0.029 cos 2Z 



Total, + 2.877sinZ 2.071 cos? 0.014sin2Z 0.051 cos2Z (a) 



2. In the logarithm of radius vector. 



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

 Saturn, 34 sin I 

 Jupiter, 11 sin I 51 cos I 



Total, 44 sin Z 37 cos Z (Z>) 







Changes in the functions e sin 7t and e cos n, represented by 5/t and 8k, will pro- 

 duce the following changes in the longitude and log r, 



to = 2 Sfc sin Z 2 Mi cos Z + f (We 7tM) sin 2 Z | (7^7i + Jfik) cos 2 Z 

 <5 log r M /i sin Z M &k cos Z. 



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

 shall vanish, we must put 



M = + 1".03G ; & = + 1".438, 

 which will produce the inequalities 



to = + 2".877 sin Z 2".071 cos Z + 0".007 sin 2 Z 0".037 cos 2 Z 

 6 log r 21 sin Z 30 cos Z. 



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



to = 0".021 sin 2 I 0".014 cos 2 Z 

 <51ogr 23 sin Z 7 cos Z. 



So that the constants of /* etc. are 



Constant of P sl 



P*= 



P s2 -_0".021 

 P c ,= 0.014 

 #,! = 23 

 3a = - 7 



