112 
PROFESSOR DONKIN ON THE 
(see equation (47.))? and the final equations become, after simple reductions, 
« « cos i + /3 cosj , * , \ 1 
0 -~ k +A^+ r ) 
?-i=f-(s-c)*cosi-e+r) 
'P- 
T~- 
J ~k 
(R.) 
These equations comprise a normal solution of the problem. The first gives imme- 
diately 
a cos i -+ (3 cos A 
cos 0= cos i cos^y — sin i sinj cos ^(£+r) 
(see art. 40.) ; and since I, J are given explicit functions of 6, the three variables 
d, <p, are determined explicitly as functions of t. The third equation (R.) simply 
expresses that the invariable plane intersects the ecliptic in a fixed line, whose longi- 
tude is j- 
45. Let us now introduce the supposition that A and B are unequal, and that the 
body is acted on by disturbing forces. 
We must (see art. 41.) put instead of ^ in the equations (R.) of the last 
article ; these equations will express the solution of the problem, the elements being 
now variable, and determined as functions of t by the system of equations 
*=-§’ («»«'=-£ (cos j)'=-% 
r =- 
d<C 
, d<P 
a = — j 
da. 
dh ' d cos i d cos j 
where <£> is the disturbing function, expressed in terms of the elements and t. 
46. If there are no disturbing forces, O reduces itself simply to Cl (art. 41.), which 
is now to be transformed by means of the equations (R.), art. 44, as follows. 
Since v=kcosj, and w=k cos i, we have 
v cos 0 — w , cos i — cosy cos f 
sin I 
sin ( 
•k sinj cos I. 
Also the expression for u, art. 42, is easily put in the following form : 
u- 
— — ^ sin 2 i sin 2 j— (cos 6— cos i cos j)‘ 
sin 1 
k _ 
= “smS SiniSin ^ Sm0 
(with respect to the sign, see art. 42.). And since 
sin ® sin I 
sin 0 sin i ’ 
u= — ksinj sin I. 
this becomes 
