1896.] Earth's Free Eiderian Precession. 185 



Thus the angular velocity of free precession, as seen by an 

 observer partaking of the motion of the solid 1 , bears the ratio 

 (C — A)/ A sect to the angular velocity of the rotation: when the 

 amplitude of the precession is small, the ratio reduces to (C — A)/A. 



3. We have now to examine in what respects this argument 

 is modified when there is elastic yielding to the centrifugal force 

 of rotation, instead of absolute rigidity. As the period of relaxa- 

 tion of a strain in the Earth is a fraction of a day, while that of 

 the Eulerian precession is measured in hundreds of days, it follows 

 that the distortion due to centrifugal force will follow with great 

 exactness the movements of the instantaneous axis of rotation : 

 and the same will usually hold good in other problems of slow 

 and small free precessional motion. Thus if C and A' represent 

 the moments of inertia, round OC and OA, of the configuration 

 which the body would assume when the strain due to centrifugal 

 force is removed, — or, what is the same thing, of the configuration 

 it would assume under the action of an applied bodily forcive 

 equal and opposite to the centrifugal force of the rotation — the 

 angular momentum will now be made up of the components, 



C'w cos i round OC, 



— A'o) sin c round OA, 



Ico round 01, 



where / is the increase of moment of inertia round 01 due to the 

 centrifugal bulging. 



Thus the reacting couples, in the plane COT, are 



Ceo cos i . w sin 7, 



— A' co sin i . to' cos y, 



Ico . &>' sin (7 — 1) ; 



and these must balance. When the axis 01 does not deviate far 

 from OC, the moment of inertia round it is equal, to the second 

 order of small quantities, to that round OC ; thus I = C — C , and 

 the condition for steady precession is therefore 



C tan 7 + (C — C) (tan 7 — tan c) = A' tan 1, 



that is, Ctan7={(7-(C"-yl 1 )} tan t ; 



so that, by the geometrical relation (1) 



n = C-(C'*A') C0Sl < 4 >* 



1 This is different from the precession of OC or 01 in space, which is round the 

 fixed axis OL, and of angular velocity 0', where !2'sin7 = « sint. 



