THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 113 
The expression for y is 
vy = IV + periodic terms. 
. 
; h ; ; ; Ii 
The approximate value of ie being 1, the complete expression for the integral of d 7 
U 1 
is given by 
hy 
1 = 1 + k; + periodic terms, 
k; being the constant of integration. 
2 = B h h zs ees 3 
Putting (3»* — 4r° + ete. = — 2» C — 1) = JV,-+ periodic terms, and substi- 
U U 
s A 6 F lt > 6 : dz 
tuting this expression, together with those of » and , in the expression for uw? 
U 
have, preserving only the constant terms, 
I = y, (k; —k —4— %, se Vad 
It is necessary now to find the value of &; in terms of the constants. If in the 
7 Gai F ; : 9 9 
expression for ae given by equation (18) we write for p , its equivalent a cos “o 
— & p cos w , we will have 
dW, = hy} 2° co (f—0) 1-2, ee Oe) Ty es ‘e) dt 
hg! di, COS *g ° hy? a, cos*y, J \df 
+ 2h) p sin (f —o) ( \dt. 
We also have 
a =h =) dt. 
Selecting from the expression for dJV, the terms not containing p cos o and 
p sin @, we have 
dQ 
dW,=—h, (1 +27.) oe 
A. P. S.—VOL. XIX. O. 
) dt. 
