DIFFERENTIAL EQUATIONS OF DYNAMICS, ETC. 
107 
easily obtain 
sin 2 ^ — cos 2 
« + /3 
2 
whence it is plain that x= cos («+&), and in like manner may the equation (41.) be 
established. 
40. Returning now to the problem of rotation, and supposing, for convenience, 
that the question refers to the motion of the earth about its centre of gravity, the 
following will be the signification of the symbols employed. 
A, B, C are the moments of inertia about the principal axes of the earth, viz. the 
axes of x,y, z; the last being the polar axis, and the arrangement being such that 
the positive direction of z is to the north pole, and that the positive axis of x follows 
that of y in the actual rotation about the polar axis : p, q,r being the angular velo- 
cities about the three principal axes, the usual convention will be adopted as to their 
signs ; so that in the actual case r is positive. The arrangement of the fixed axes of 
ii, £ is supposed similar to that of x, y, z, the plane of |, being a fixed ecliptic, and 
the axis of | the origin of longitudes unless another origin be expressly indicated. 
Then 0 is the oliquity, 4 the longitude of the vernal equinox, and <p the right 
ascension of the axis of x ; all referring to the fixed ecliptic. 
Let the <f principal plane” signify that which, in the undisturbed problem, is the 
“ invariable plane.” Then i is the inclination of the principal plane to the fixed 
ecliptic, andy is the inclination of the equator to the principal plane. 
In the case of the earth, A is nearly equal to B, 0 never differs sensibly from i, and 
j is therefore always small. But these conditions are not supposed in what follows. 
It is assumed however that C is the greatest of the three moments of inertia. These 
conventions, in which it is very desirable to avoid any ambiguity, may be illustrated 
by the annexed figure, in which O represents the origin of longitudes. 
The angles of the spherical triangle formed by the intersection of the three planes 
with a spherical surface are i,j, ir—0; and the sides opposite to them will be denoted 
by I, J, 0. Thus we shall have 
cosl = J - ■> nos. I 
j cos i — cosy cos 0 t cos j — cos i cos 6 
cos 0= cos i cos j— sin i sinj cos 0. 
p 2 
