GEOLOGICAL CHANGES ON THE EARTH’S AXIS OF ROTATION. 
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2 ( 0 , t. Then reduce the earth to rest, and during the next interval let the matter 
constituting the earth flow (with velocities double those with which it actually flows) so 
that the pairs of principal axes have, at the end of the interval, rotated with respect to 
the third ones through the angles —2 ar, —2/3 r, —2 yr. Lastly let 2# i r, 2 0 2 r, 2 $ 3 r be 
the rotations of each pair of axes about the third by which they could have been 
brought directly from their initial to their final positions in the time 2r. 
Therefore, by the principle of superposition of small motions, 
Q 1 —a> 1 — a, Q 2 =u 2 — (3, Q 3 =cj 3 — y. 
Now supposing these two processes to go on simultaneously with their actual velocities, 
instead of in alternate intervals of time with double velocities, it is clear that Q L , 6 2 , Q 3 
are “the angular velocities of the axes with reference to themselves”; a lt u 2 , a 3 are the 
component angular velocities of the earth considered as a rigid body ; and —a, — 0, — y 
are the component angular velocities of the principal axes relatively to the earth, arising 
from the supposing continuous distortion of that body. 
With respect to the other quantities involved in the equations of motion : — 
Let C, A be the principal moments of inertia of the earth initially when t is zero ; 
and at any time t, let 
?q=A+a£, X 2 =A+b£, X 3 = C+c7. 
We here suppose that the changes in the earth are so slow that terms depending on 
higher powers of t may be neglected. 
Lastly the quantities H 15 H 2 , H 3 are respectively twice the areas conserved on the 
planes of 0A, 0A by the motion of the earth relative to these axes. If the earth 
were rigid, they would all be zero, because there would be no motion relative to the 
principal axes ; thus <y n a> 2 , u 3 do not enter into these quantities. Now the motion which 
does take place may be analyzed into two parts. Divide the time into a number of 
equal small elements r, and in the first of them let the matter constituting the earth 
flow (with a velocity double that with which it actually flows) ; this motion will 
conserve double-areas on the planes of Q 2 Q 3 , 0&, which we may call 2phr, 2f| 2 r, 2^ 3 r. 
In the next interval of time let each pair of axes rotate round the third (with angular 
velocities double those with which they actually rotate), so that at the end of the 
interval they have turned through the angles —2 ar, — 20r, —2 yr. Now since during 
this second interval the axes have rotated in a negative direction through the solid, 
therefore the solid has rotated in a positive direction with reference to the axes. 
Remembering then that a 1? X 2 , X 3 are the principal moments of inertia, the double-areas 
conserved on the three planes in this second interval are 2\ut, 2x 2 (3t, 2 x 3 yr. Hence if 
2H 1 t, 2H 2 t, 2H 3 r be the double areas conserved in this double interval of time, we have 
2H 1 r=2l J r+2^ 1 ar, 2H 2 r=2^ 2 r+2\Pr, 2H 3 r=2l 3 r+2 X 3 yr. 
Therefore if we now suppose the two processes to go on simultaneously with their actual 
velocities, instead of in alternate elements of time with double velocities, and if we 
substitute for 7q, X 2 , their values in terms of A, a, t , &c., we get 
