OF THE ^PERTURBATIONS OF THE PLANETS. 
219 
JR 
(d R 
\ dr 
_/ d R N 
v r 
t - _ • 
da ~ 
Ur 
) da~ 
” V d r t 
' a ’ 
d R 
(d R 
\ dr 
R + 1 
d v 
de ~ 
XTr 
) ' di ' 
* 
~dl’ 
d R_ 
de 
(d R 
\ dr 
) • (£ 
+ R- 
de) + 
d R /d R\ (dr dr w \ (d R\ dv 
d m \ dr ) \d m' dv dm) ' \dv) * dm’ 
(C) 
in which expressions, it need hardly be observed, that ^y, jy, refer to 
this value of v, 
v = £ + s — d>. 
Proceeding 1 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, 
1 1 „ /■>-,,d a 
a 
== ^ — 2 J*d) R: consequently, = 2 d R. 
Now, when x, y, z are transformed ifito 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 R will be 
found by differentiating R, making £ the only variable, that is, we shall have, 
dR = ^dx+ -j^dy + ^7 dz = 
dx _r dy 
But substituting this value, 
da = 2a 2 d£. 
d$ 
a 
= a + 2ftf dZ,\ 
( 12 ) 
The mean motion £ is defined by this equation, d £ = 
dt \/fj. 
a- 1 
But, we have, 
f = 10 + 2 a / d ' A ) '• aad - dJ ^ = ^ • (i + 2 ■■/*»)*• 
aJ 
2 F 2 
