THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 111 
aw 
— we have 
dy 
Then in case of — 
tol 
+ 3k, sin y — 3k, cosy + 7 k, sin 2 y — » kh, cos 2y + ete. 
In the second integration we call the two new constants Cand J, and the con- 
stants of the results are in the forms 
C+ kh nt + k sing —k, cosg + $7 k, sin 29 — $ 7° kh, cos 2g + ete. 
ol— 
Nf —4k,cosg— tk, sing — $7” k, cos 2g — $7 hk, sin 2 g — ete. 
In case of the latitude the constants are given in the form 
AQ +isng+l,cosg + 7 1, sin 2g + 7° L cos 2g + ete. 
The constants are so determined that the perturbations become zero for the epoch 
of the elements. Hence also the first differential coefficients of the perturbations 
relative to the time are zero. We substitute the values of g and g’ at the epoch in 
4 u d : a ; 9 
the expressions for ndz, », — , oa (ndz), ete., including in g’ the long period term. 
cost” na 
Putting the constants equal to zero, and designating the values of néz, v, ete, at 
the epoch by a subscript zero, we have the following equations for determining the 
values of the constants of integration: 
C+ ksin g —k, cosg + $7 k, sin 2g — $ 4k, cos 29 JL ef, at (ndz)y = gy 
= 7 Gh, ; 
k + k,cosg + &sing + 7° k, cos 2g + 7° k, sin 29g + ete. + — (nbz) = 0 
ndt 
N — 1k, cosg—ik, sin g— $7 k, cos 2g — 4 7 & sin 2g — ete. + GO); = 0 
2 9 1 
+ dk, sin g— tk,cosg+ 7° hk, sin 2g — 7 k, cos 2g + ete. + a (7) = 0 
i+ %sn g+becosg + 7” 1, sin 2g + 7° L cos 2g + ete. 4 i= Jo = (I 
d ni 
1, cos g — l, sin Gs 7” 1, cos 2g — 7 1, sin 29 + ete. + ( : y. 0) 
ndt \eost 
