DIFFERENTIAL EQUATIONS OF DYNAMICS, ETC. 
109 
from which the following also are easily deduced : 
u = — ^ - ^ '+ A 0 B (^' cos2<p+4/ sin0sin2<p) 
V=C (p' + d/ COS 0) 
| I ^ 
w = — — %// sin 2 4+C cos 6 (<p ' cos — o — sin ^ sm 2< p—4 1 ' sin ^ cos 2 <p). 
41. Resuming the three integrals (i.), (ii.), (iii.) of art. 31, we may put the first in 
the following form : — 
Z-\-Cl=h, (i.) 
(vcos 0 — w) 2 \ t V 2 
C 
i • i rw 1/1 , 1\ / , , (VCOS 9— wf\ V 2 
m which 2Z=2 U+b) ( m + ) +C 
2H=i('4- 1 
V? 
(v cos S — w) c ‘ 
sin 2 S 
’2 \ A B 
and the other two are, as before, 
(v — w cos fl) 2 
^ cos 2 <p ■ 
2m(vcos5— mj) . „ 
- sin 2 ip 1 ; 
sin ( 
tt> 2 +w 2 -l- 
sirr 
=k 2 
w=g, 
(ii.) 
(iii.) 
in which k is the sum of areas on the invariable plane, and g the sum on fixed ecliptic ; 
moreover v=Cr is the sum of areas projected on the plane of the equator; hence 
we have 
g=k cos i, v=kcosj. 
It has been seen that the complete solution of the problem is impracticable in the 
general case, on account of an algebraical difficulty. If however we suppose B=A, 
this difficulty disappears; and after completing the solution on this supposition we 
may take account of the terms arising from the inequality of A and B, by treating 
the function denoted above by fi (equation (i.)) as a disturbing function, and apply- 
ing the method explained in art. 38. Thus when the action of disturbing forces is 
considered, the whole disturbing function will consist of two parts ; one depending 
upon the forces, and the other the function which has just been assigned, and of 
which the effect, as will be seen, is extremely simple. 
42. We proceed then first to complete the solution on the supposition A=B. The 
three integrals (i.), (ii.), (iii.) give in this case 
^ 2 =c~a(^— 2A ^)j W ~S ( 44 *) 
u= ^^\k 2 —v 2 —w 2 -\-2vw cos 0—k 2 cos 2 dj*, 
in which latter expression the above constant values of v and w are to be introduced. 
We may put, as before, g=k cos i, v=k cos j,j being now constant ; and the expres- 
sion for u becomes 
M ~sin"0i l— cos 2 *— cos^+Scosicosjcos^— cos 2 0i . . . (45.) 
