THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 107 
and 
a=—sinzsino, (6 =sin7zcoso. 
The values of p and qg then are given by the equations 
p= —a cos 2, — sin Qo, 
g = — asin 2 + 6 cos Q — sin 
from which we have 
dp da. dp 
a = — GCOS Qo Fh SIN {2 Fe 
dq 3 da dp 
Ros SIN {2 as + GOS a: 
From the equation z, = a « + @ y we have, first regarding a and ( as constant, 
then regarding # and y as constant, 
dx 5 dy 
Giseetes 
dz, da. Ap oe 
al = 0 FF SP I) ap SO 
Differentiating the first of these, regarding all the quantities variable, we have 
az da dx dp dy au p GY 
= oa 
dt’ dt dt dt dt dt? dt? 
Z, being the component of the disturbing force parallel to the axis 2, and X and 
Y the other two components, we have 
Z=aX+ 6 V+ Zcosz. 
Writing for X and Y their values 
vx 
di? 
ay 
+h (1+m)”, ated +m)s, 
