GEOLOGICAL CHANGES ON THE EARTH’S AXIS OE ROTATION. 
309 
and let A, B, C be the principal moments of inertia at that instant. Let h be the 
constant moment of momentum (or twice the area conserved on the invariable plane). 
Consider axes fixed relatively to the solid in the positions of the principal axes at any 
instant, but not moving with them, if they are being shifted in virtue of changes in the 
distribution of portions of the solid. 
The component angular velocities of the rest of the universe are, relatively to these 
axes, ; and therefore, if N be the point in which the normal to the invariable 
plane at the origin cuts a sphere of unit radius, the components parallel to these axes 
of the velocity of N relatively to them are 
Now, suppose that by slow continuous erosion and deposition the positions of the 
principal axes change slowly and continuously relatively to the solid. 
Let vs, £, a be the components round the axes (which, of course, are always mutually 
at right angles) of the angular velocity of the actual solid relatively to an ideal solid 
moving with the principal axesf. Then the component velocities relatively to this ideal 
solid of the point of the body coinciding at any instant with N are 
zp — ya , xs — zvs , yvs — xg ; 
and the components parallel to the principal axes of the velocity of N relatively to these 
axes are Hence we have 
These three equations give 0, and therefore they are equivalent to 
two independent equations to determine two of the three unknown quantities x, y , z as 
functions of t, the three fulfilling the condition x 2 -\-y <2 -\-z 2 = 1, and it being understood 
that vs, g, a are given functions of the time. 
To apply these equations to the questions proposed as to the earth’s axis, let the 
normal to the invariable plane be very nearly coincident with the axis of greatest moment 
* The angular velocity of the rest of the universe relatively to the earth being opposite to the angular velo- 
city of the earth relatively to the rest of the universe, the components of the former round the axes x, y, z are 
taken as in the negative direction, i. e. from z to y, x to z, y to x. 
f w, p, a are the same as — a, ~/3, —y of Part I. 
j These equations are the same as those given by me in Part I. p. 277. 
