OF THE PERTURBATIONS OF THE PLANETS. 
201 
orbit which the planet describes in the time d t. The last equation may there- 
fore be thus written. 
d r 2 r 2 d v 2 
ft. dt 2 ' ft. dt 2 
- — + — = 2fd'R. 
- r 1 a J 
(2) 
The double of the small area contained between the radii r and r -f- dr, is 
equal to r 2 dv ; and as x, y , % and x + d x, y -f- dy, z + dz, are the coordi- 
nates of the extremities of the radii, the projections of the area upon the planes 
of x y, xz, y z, are respectively equal to 
xdy—ydi v, xdz — zdx, y dz — z dy : 
wherefore, according to a well known property, we shall have, 
r* dv 2 ( x dy — yd x ) 2 (x dz — z d x ) 3 [y dz — z dy ) 2 
ft. dt 2 ft. dt 2 \t. dt 2 ft. dt 2 
and the differential of this equation, d t being constant, may be thus written, 
, r* dv 2 n , a . o i (dxddx + dyddy + dzddz 
= + fjf- 
- 2 (xd* +ydy + zdz) . ( ««* + + 
Now, substitute the values of the second differentials taken from the equations 
(A), and we shall obtain, first. 
dxddx + dyddy + dzddz dRj , dK^ , 
VdLf- — ~dx Cl X + lly d y + 
d R 
d z — ^ — d! R — 
d r 
ft.dt 2 
and, secondly, 
x d dx + y d dy + z d d z d R d R d R 1 d R 1 
= i X x + Xjy + -iu z ~T = dV r -T : 
wherefore, since x 2 + y 1 + z 2 = r 2 and xdx-{-ydy-\-zdz = r dr, the fore- 
going differential equation will become by substitution, 
r* d v 2 ( . d R 7 \ 
d -JJ¥ =2r2(rfR- -jjdr), 
or, which is equivalent, 
r* d if 
d. 
V- 
dt 2 
0 (d R , d R , 
= 2r ‘{^dK+ — ds 
ds 
)• 
2 D 
MDCCCXXXII. 
