348 
PROFESSOR DONKIN ON THE 
would be given functions of t. Then if we call 
J the inclination of the principal plane to an arbitrary fixed plane ; 
Q the longitude of the line of intersection of these two planes, reckoned in the 
fixed plane from a fixed line ; 
N the angle between this line of intersection and the line of nodes', 
we should have (as in art, 83. with a different notation) 
cos N — sin N 1 
Q' sin sin N+^yi cos N I . . (118.) 
N'=ii; 2 — cot J(a»o sin N+iy, cos N)J 
and the integration of this system would give J, Q, N as functions of t, and so deter- 
mine at any instant the position of the principal plane and of the line of nodes, rela- 
tively to fixed space. 
With respect to the motion of the principal plane, the following may be added. 
It has already been shown (art. 88, 89.) that the line about which it is at any instant 
turning, coincides with that in which it is intersected by the plane of the radii vectores ; 
and the values of Uq, (see art. 88, putting ^=0 in the expressions there given) may 
be put in the form 
sin I. (jo cos 6 sin sin 6 cos 
oM^ — mni^ sin If ^ ) sin ^sin 
If the latter of these be multiplied bye, and then both sides of each squared, and the 
results added (after putting for a- on the right its value f-^'2pp^ cos 1), we find, 
observing that cos ^cos^^-f- sin ^ sin cos 1= cos%, 
e"\/ col-\-a)\=mm^ sin I (p^ sin^ ^^-l-/>^sin^ 0-\-2pp, sin ^ sin 6^ cos 
an expression which may be further transformed as follows. Let X, be the latitudes 
of m, ni^ (with reference to the principal plane) ; then sin X— sin ^ sing sinX^ = sin^, sin/, ; 
, • trsiiif, — (T sin » • 
hence, since p = ~ — p,= — ^ ^ ? we obtain 
^ sin 1 ^ ' sm 1 
(r^/ (sin" X-\- sin" >.,—2 sin X sin X, cos %)i 
This gives the absolute angular velocity with which the principal plane is at any 
instant changing its direction in space ; it is evident that (if the supposition r=r be 
excluded) it can never vanish except when both planets are in the line of nodes. The 
direction of the rotation is determined by the signs of and 
96. The system of differential equations given in art. 93. affords an example of the 
so-called “elimination of the nodes” in the problem of three bodies, Jacobi, by a 
very remarkable and ingenious transformation, has effected the elimination in a quite 
