OF THE PERTURBATIONS OF THE PLANETS. 
199 
moves in the circle will be proportional to yj [x. Thus the algebraic quantities 
t y/f* and d t yj p 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 A. and X 1 denote 
the longitudes of the planets P and P / 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 
x' = 
r 1 cos A' 
■ v /1 +S 25 
a /1 + s'*’ 
y = 
r sin A 
y= 
r' sin A' 
\/ 1 + s*’ 
x/1 + s' 2 ’ 
z = 
r s 
z' = 
r' d 
X/I +5 2 ’ 
x/1 + s' 2 ' 
In the transformations alluded to, the quantities 
d R d R d R . 
-3J> SX 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. 
rfR d R dr d R d A d R d s 
dx dr ‘ dx ' d \ ’ dx ds ’ dx* 
and having computed the differentials j~ x from the formulas 
r = yj ' x 2 + y 2 + z 2 , tan . X 
_ y 
s = 
a/ x 2 + y~ 
d R 
the substitution of the results will make known the expression of By the 
like procedure the values of ^ and ^7 will be found 
d R 
d R 
cos A d R 
sin A a/ 1 + s- 
dR 
cos A s s/ 1 + s 2 
dx 
d r ’ 
a/1 + s 9 dA 
r 
ds 
r 5 
d R 
dR 
sin A , d R 
COS A a/ 1 + S 2 
dR 
sin A s a/ 1 -(- s 2 
dy ~ 
dr ’ 
x/1 + s 9 d A * 
r 
ds 
r 5 
dR _ 
d R 
s 
+ 
d R 
a/1 + s 3 
d z ~ 
d r ’ 
a/I + S 2 
ds 
r J 
>• • • (B) 
