OF THE PERTURBATIONS OF THE PLANETS. 
219 
d R / d R\ dr / d R\ r 
d a ~ \ d r ) d a \dr ) a 5 
dR /^R\ dr dv . (d R\ dv 
d s xdr/'dv ds^xdv/'de’ ' 1 
' *7R /^R\ (dr dr dv\ /c?R\ dv 
de \ dr ) ’ ' dv de) ' \dv ) de ’ 
d R / d R\ (dr dr dv\ , /d R\ d v 
dzr \ dr ) dv d^J ' \ dv ) ’ diz' > 
in which expressions, it need hardly be observed, that refer to 
this value of v , 
v = £ + s — c b. 
Proceeding now to reduce the differentials of the elements of the variable 
orbit to the forms best adapted for use, we have this formula for the mean 
distance a, 
— 2 \Ji£ R : consequently, ^ = 2 d R. 
Now, when x, y, z are transformed into expressions of £ and the elements 
of the orbit, it has been proved that dx, dy, dz contain d £ only, and are inde- 
pendent of the differentials of the elements : wherefore, the value d' R will be 
found by differentiating R, making £ the only variable, that is, we shall have, 
C?R 
<7R 
rf ' R = ~d^ dx + 7R7 d V + di dz = 
dy 
d R , „ 
But substituting this value, 
The mean motion £ is defined by this equation, d^=z dt vA But, we have, 
T = } (1 + 2 a/d' R) ; and, ^ . (1 + 2 a/d' R)* 
2 f 2 
