224 
MR. IVORY ON THE THEORY 
= — r sin ( [v — P) cos N sin i = — z cos N, 
. rsin(X — N) 
—~ = r sin (v — P) cos i = — j 3 . 
di v ' V 1 + S 
. .11 c/R JR rfR , , 
Let these expressions be multiplied respectively by -jp and then 
added ; the result will be 
JR 
d i 
U4^sinN-|5 C o S N} + ^.^ 
i \ dx dy J dz V 1 
(x — N) 
+ s 3 
and by substituting the values contained in the formulas (B), 
d R / ■ n\ d R • . T\ ^ R .. \t\ 
■JJ = (1 + s ) 77 sm (^-N) - ^s cos (X -N): 
d S a 
and, because s cos (X — N) = sin (X — N), 
d R ... XT . f,i I o\dR JR ds\ 
77 =sm(N-N). ((!+«*) 77 - 77 ' 5 * j • 
R d R d R 
If the equations (11) be multiplied respectively by jj, -jp and then 
added, this result will be obtained, 
d R j • - d R j » ._ _ 
-dT dl + 7TK d N = °- 
d N 
d R 
By combining this equation and the value of with the formulas (6), we get 
JN = 
di = 
JR 
a/ i _ e 3 sin i d i 
f- d L 
1 
a 1 d R 
a/ i — e 1 " sin i * d N 
■ d l t 
J 
( 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 Z,. 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 
