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, 
+ --I + V = 
dr 2 
jxdt 2 ' [xdt 2 
( 2 ) 
The double of the small area contained between the radii r and r dr, is 
equal to r 2 dv ; and as x, y, z and x -f- dx, y -f- dy, z + dz, are the coordi¬ 
nates of the extremities of the radii, the projections of the area upon the planes 
of xy, x z, y z, are respectively equal to 
xdy — ydx, xdz — zdx, ydz — zdy: 
wherefore, according' to a well known property, we shall have, 
r*dv 2 {xdy—ydx) 2 (xdz — z 
ydt°~ .dt 2 ft .dt 2 
dx)~ t {ydz—zdy) 2 
H— 
/x d t 2 
and the differential of this equation, dt being - constant, may be thus written, 
, r*dv 2 _ . „ . „ . (dxdd.x + dyddy+dzddz\ 
d.—n*= 2 y? + y* + %*) . (- fj/- -) 
dt 2 
7 . 7 . J \ (xddx + yddy + zddz\ 
2(xdx + ydy + zdz) . --- ) 
Now, substitute the values of the second differentials taken from the equations 
(A), and we shall obtain, first. 
. d R , d R dr . dr 
x ^~d^ d y^r^rz dz — ^ — d!R- 
dx ddx + dy d dy + dz ddz _ d R , 
[x d t 2 d x 
and, secondly, 
xddx+yddy + zddz d R dR dR 
fxdt 2 dx X ' dy y ' dz 
wherefore, since x 2 + y 2 z 2 = r 2 and xdx-\-ydy-^zdz = r dr, the fore¬ 
going differential equation will become by substitution, 
dz 
1 d R 1 
r dr 1 r 
r* dv 2 d R 7 \ 
d -Jdf =-2rZ(dll- T7 dr), 
or, which is equivalent. 
7 ridw 2 9 /dR . , dR , \ 
d.-^ = 2ry jJ ;dX + Y7 ds). 
[xdt 2 
2 D 
MDCCCXXXII. 
