224 
MR. IVORY ON THE THEORY 
dv r ? . 
= — r sin ( v — P) cos N sin i — — z cos N, 
dz . , ™ . rsin(A — N) 
-j-. = r sin (y — P) cos i = — j . 
d i v ' V l + s a 
, . , 1 . T 1 . . . ^P *ZR ^P 1 1 
Let these expressions be multiplied respectively by ~rj, and taen 
added ; the result will be 
d R 
d i 
f d R • XT d R XT *) , d P 
= z < — Z sin N cos N > + -r- . 
I dx dy J dz 
< 7 R rsin (X — N) 
>/ l + s 2 
and by substituting the values contained in the formulas (B), 
77 = ^ + * 2 ) ^ s!n “ XX * cos <v -N) : 
d s 
and, because s cos (X — N) = sin (X — N), 
^S = sin(X-N) . {(1+* 2 )^- 
d R ds 
~d~k ' d~k 
}• 
If the equations (11) be multiplied respectively by and then 
added, this result will be obtained, 
d R 7 . * d R j » . A 
<7R 
By combining this equation and the value of -jj with the formulas (6), we get. 
d N = 
di = 
1 <?R 1 
sin i * di ’ 
1 <7R 
sin i ’ d N * 
(18) 
The differentials of the several elements of the orbit of the disturbed planet 
have now been made to depend upon the function R and its differentials rela- 
tively to the elements themselves and to the mean motion Upon the cal- 
culations which this transformation requires, which have long ago been car- 
ried as far as human perseverance can well be supposed to go, we do not here 
enter. The variations of the elements of a disturbed planet, in the most per- 
fect form in which they have been exhibited in the latter part of this paper, are 
the result of the repeated labours of Lagrange and Laplace, who, at different 
