OF THE PERTURBATIONS OF THE PLANETS. 
221 
the semi-parameter of the primitive ellipse being equal to a (1 —e 2 ), and its 
eccentricity to e'. 
The eccentricity is determined by this formula, 
2 . & 
e =1 ~ 
by differentiating, 
7 hdh , h 3 d a hdli . ,, d R , 0 
ede=-~ + T . rfT=- — + 
a 1 2 * a* a 
and by substituting the value of h d h, 
de 
— — a^l — e 2 . |i 
— a/ i 
rZ R d R 
+ 
* dX, ^ edrs 
]•<*& • • • (15) 
For the variation of the perihelion \ve have the formula (7), which may be 
written in this manner. 
2 
— hdh = cos (v — rs) de + e sin (v — w) dzs\ 
and by multiplying all the terms by e , 
0 ^ 
e sin (v — 7s) . edzs = ^ hdh — cos {y — Ts)ede\ 
and because ede— — + h 2 e?' R, 
esin (v — rs) .edvs — (y- -f- C ° S V -—— J hdh — h 2 cos (w — w) d' R. 
Further, rf'R=^+ (|£) (fjl)..***: 
and, by substituting this value, 
. , N , /2e , cos (v — ct) Id cos (u — *ar)\ 7 77 
e sin (v — vs) . ed zj = f — +----p-) . hdh 
hr dr (d R\ „ , 
— r 5 " d5 cos (*~"’) (dr) r2dt ’- 
dr 
Now -3 . = esin (v — zr ); and it will be found that the coefficient of h dh is 
equal to, 
(2 + e cos (v — . e sin 2 (u — w) 
1 dv 
