OF THE PERTURBATIONS OF THE PLANETS. 
199 
moves in the circle will be proportional to Thus the algebraic quantities 
t y/[jj and d t [m represent the arcs of this circular orbit, which are described 
in the times t and d t. 
It is requisite in what follows to transform the coordinates x, y, z into other 
variable quantities better adapted for use in astronomy. Let X and X 1 denote 
the longitudes of the planets P and P 7 reckoned in the fixt plane of x y, and 
s and s' the tangents of their latitudes, that is, of the angles which the radii 
vectores r and r' make with the same plane: then, 
X = 
r cos A 
11 
r 1 cos A f 
V 1 4- V 5 
V 1 + s n ’ 
y = 
r sin A 
II 
r’ sin X 1 
V 1 + s~ 
V'l + s' 9 ’ 
z = 
r s 
z r — 
r 1 sf 
V'l + s 13 ' 
In the transformations alluded to, the quantities -jj must be ex¬ 
pressed in the partial differentials of R relatively to the new variables r, X, s ; 
and it will conduce to clearness of method if these calculations be dispatched 
here. We have the equation, 
d R 
dx 
cZR dr d R dx d R 
dr dx ' dx ’ dx ' ds 
d s 
dx ’ 
d p d> ^ d s 
and having computed the differentials from the formulas 
= + y 2 + 
tan . X = 
— y. 
X 
V* 2 + y°~ 
the substitution of the results will make known the expression of 
like procedure the values of and 4— will be found 
JR 
d x' 
d z 
d R 
dx 
dR 
dy ' 
dR 
d z 
d R 
d r 
dR 
d r 
dR 
d r 
cos X 
\/l + s 3 
sin X 
dR sin A V1 + s 2 d R cos As VI + s 2 
dx r ~ ds r ’ 
v'l + s 2 
5 
, d R cos X V1 + s 2 
+ Tx • 7 
\Xi + s 2 
d R sin A s V1 -I- s 3 
ds r '■ 
, f[R v'l + s 3 
+ ds ' ~~~ r ‘ 
By the 
> 
(B) 
