OF THE PERTURBATIONS OF THE PLANETS. 
215 
5. The expressions of the coordinates x, y, z, in terms of the variables 
r, X, s, are as follows: 
x = 
r cos A 
^ i + s - 5 y v i+s 3 
and, if we write X — N + N for X, we shall get. 
r sm x 
2 = 
r s 
+ s 3 
C cos (x 
x = r ‘l~7T 
cos (x — N) 
y 
f si n (A. 
= r ' { 
+ S 2 
sin (x — N) 
cos N — 
sin (x — N) . 
+ s 3 
sinN 
+ S 
XT , cos (x — N) . xT 
2 cos N d- ;-p— ■ sin N 
■ 3 V l + s 
}> 
}' 
But v — P in the plane of the planet’s motion is the hypothenuse of a right- 
angled spherical triangle of which X — N is one side, 5 the tangent of the 
other side, and i the angle opposite to this latter side ; and from these consi¬ 
derations we get 
cos (X — N) 
</F+l r 
= cos (v — P), 
sin (X — N) . 
—; v = sin (v — P) cos i, and 
a/ 1 + s 3 v ’ 
a/ 1 + s 3 
= sin (v 
P) sin i: 
wherefore we have these values of the coordinates, 
x = r . (cos (v — P) cos N — sin (v — P) sin N cos *} 
y — r . {sin (y — P) cos N cos i + cos {v — P) sin N} 
z = r . sin (v — P) sin i. 
The radius vector r is a function of v, a, e, vs, viz. 
a (1 — e 2 ) 
ty* ■■ ■ . . v —_- • 
1 + e cos (u — ot) ' 
and thus the coordinates x, y, z, are functions of v and the five elements a, e, 
tar, N, i ; for P is no independent quantity, since it varies with N. In order to 
abridge we may write X, Y, Z for the multipliers of r in the foregoing expres¬ 
sions of x, y, z ; so that 
x — r X X, y = r X Y, x = rxZ. 
Now, on account of the equation (8) we have 
