THE GENERAL PERTURBATIONS OF TITLE MINOR PLANETS. 15 
[f we now put 
vsin V= 2a.cos¢.ksin (II— KX) 
vcos V= 2a.cos.cos 9’. k, cos (Il — K,) 
wsin W = p— 2a’. nee 
weos W= v.cos(V— P) 
w,sin W,= v.sin( V— P) 
w, cos W,= 2a’. a cos P, 
| 
(10) 
J 
we get 
J.sin(#— P)=w.sin(e + W)—ep 
f.cos(f#— P) = w,. cos (e+ IV,). (QU) 
Further, if we put 
R=1+0°?—2a’.e, (12) 
we have 
Yo = R— 2e.cose + €’. cose + ey, 
or, ¥) = R—2e.cose+e.cos*s +e .feos F. (13) 
We find the value of y, from 
The constants, *, A, hk, Ay, p, P, w, W, w,, W,, 2, are found, once for all, from 
the equations given above. For every value of « we have the corresponding value of 
Jj and F from equations (11); hence, also the values of fsin #, fcos /, which are the 
values of @) and 7, Equation (13) furnishes the value of y) by substituting in it the 
various numerical values of <, as was done for 3, and y,. ‘The value of the coefficient 
: ‘ A\2 : 
y. being constant, we thus have given the values of (“) for as many points along 
a 
the circumference as there are divisions. 
