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, 
by differentiating, 
e de = — 
h dh , h 2 d a h dh , (f R , 0 
= - — + a ( 1 - e2 )7? d i- 
a 1 2 * a' 
and by substituting the value of h d h, 
i — s/ \ — e 2 d R 
de 
= — a^/l — . | 
d R 
' dt, ' e dts 
• • • ( 15 ) 
For the variation of the perihelion we have the formula (7), which may be 
written in this manner. 
~hdh = cos (v — sr) d e -f- e sin (v — w) dvr. 
and by multiplying all the terms by e, 
£ ^ 
e sin {v — zs) .edzs — ^hdh — cos ( v — zs) e d e : 
and because e d e — — + h 2 d ' R, 
3, - 
esin (v-vy).edv= (~ + hdh — h 2 cos (v — zs) d! R. 
■p. .. »t, hdh . (d'RX , hdh 1 dr (d R\ , , 
Further, d R = — + dr=—+y>. TxJ . (jy) .r*dv: 
and, by substituting this value, 
. , . 7 /2e , cos (v — ot) h 2 cos (v — m)\ 
<?sin(y — zs) .edm = \jr+ p ) . 
- • rv cos ( v ~ ®) ( 37 ) dv - 
hdh 
h 2 dr 
Now js . = e sin {v — zs ) ; and it will be found that the coefficient of h d h is 
equal to, 
(2 + e cos ( v — ©•)) . e sin 2 (v — w) j 
f(r-7) = t • <sin (*>-*) ; 
