OF THE PERTURBATIONS OF THE PLANETS. 
223 
substitute now the value of dzs in equation (16), and that of hdh viz. 
a 2 (^) d £, then 
cos {v — zs) (dz - dzs) = — 2 r j? e • (^) d £ 
^ a (1 — e s ) cos v — 'sr ° ^ j - 
tfR 
d e 
But, as appears from the formulas (C), 
dv /d R\ d R , , . SdR\ 
di' te)=77 + acos(< ' _s,) \7z) : 
wherefore, by substituting and dividing all the terms by cos (y — m), 
dz — dzs = 
a (I — e-) d R 
e de 
dl-2r*(^)dZ, 
and by substituting the value of dzs, and observing that (^r ? .) 
obtain 
dz = a >J\—e 2 (---—) 
d R a 
d a 
—, we 
r * 
d R 7 <y ~ 9 d R J cs 
• (17) 
If the formulas (C) be multiplied, each by its own differential, and the re¬ 
spective results be addedj it will be found that the coefficients of and 
(ify) are ea °k e( l ua l t0 zer0 5 on account of the equations (8) and (10): so 
that we have, 
rfR , d R , rfR 7 rfR , 
da > + ^ de + ^ dz + 
d a 
di 
d zs 
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 ddS 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 
d x 
-jl — r sin (v — P) sin N sin i = z sin N, 
