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 : 
r cos X r sin X r s 
X y' 1 + s 9 ’ y v'l+s 3 ’ Z V'l+s 2 ' 
and, if we write X — N + N for X, we shall get, 
f cos (x — N) , T sin (x — N) 
x = r A — /T — ; cos N 7 == — g 
f^/l+s 2 V l + s 3 
y 
= r.{ 
sin (X — N) , T , cos (X — N) 
COS N + 
*/\ + s 2 
V l + s 3 
sinN 
sin N 
}> 
}• 
But v - Pin the plane of the planet’s motion is the hypothenuse of a right- 
angled spherical triangle of which X — N is one side, s 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) 
✓nr? r 
= COS ( V — P), 
sin (x - N) . 
— - - 3 ' = sm (t> — P) cos i, and 
v i "f r 
s 
\/TT7 
sin (y — P) sin i : 
wherefore we have these values of the coordinates, 
x — r . {cos ( v — P) cos N — sin (v — P) sin N cos i } 
y — r . {sin (v — P) cos N cos i + cos ( v — P) sin N} 
^ = r . sin (y — P) sin i. 
The radius vector r is a function of v, a, e, zs, viz. 
a ( l — e 2 ) 
y\ — — i • 
1 + e cos (u — ■»■) ' 
and thus the coordinates x, y, z, are functions of v and the five elements a, e, 
tz, 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 
a? = rxX, y = rX Y, z = r X Z. 
Now, on account of the equation ( 8 ) we have 
