OF THE - PERTURBATIONS OF THE PLANETS. 
223 
substitute now the value of dm m. equation ( 16 ), and that oi lidh viz. 
a 2 J 1 - e 2 (^) d £, then 
cos (v — m) (dz — dm) = — 2 rj? e - (^) d £ 
_ f a ( 1 - g*) cos v - ct _ 2 1 dR 
I e \ d e 
But, as appears from the formulas (C), 
dv /JR\ JR , N (d R\ 
Te' \dv) ~~de ~^ ac0S ( v ~ a ) = 
wherefore, by substituting and dividing all the terms by cos (v — m), 
7 7 afl-r’) (fR,„ _ / J R\ , y 
dt-d* = . (-jfjdl-. 
and by substituting the value of d and observing that ^ • y~, we 
obtain 
d z = a^/l — e 2 (■ 
1 — \/ 1 — e~\ d R 
)~d ( 17 ) 
If the formulas (C) be multiplied, each by its own differential, and the re- 
spective results be added 3 it will be found that the coefficients of (-jp) and 
(zu) aie eac ^ ec l lia l to zero, on account of the equations (8) and (10) : so 
that we have, 
JR , JR , JR . JR, 
d a + d e + -^7 a z + -mdm — 0 : 
J a 
J OT 
and this equation will serve to verify the values of d a, de, dz, dm, which 
have been separately investigated. 
It remains to examine whether the values of di and e?N already found 
(equation (6)), can be expressed similarly to the other elements. The three 
quantities N, P, i, or rather the two N and i, since P varies with N, are inde- 
pendent of r and v, and consequently of a, e, z, m: wherefore, by differen- 
tiating the expressions of x , y, z relatively to i, we shall get 
dx 
