186 Mr Larmor, On the Period of the [May 25, 



The validity of this result is also, as above stated, confined to 

 the case in which 2tt/Q, sin c is large compared with the time of 

 relaxation of elastic strain in the solid of revolution to which it 

 belongs. 



Since C — C is small compared with C and A , as the elastic 

 deformation only slightly alters the shape of the body, the third 

 of these balancing reacting couples can be neglected compared 

 with the other two. Thus the free precession, under the influence 

 of elastic yielding, ivill be the same as ivould belong to an absolutely 

 rigid body, whose configuration is the one that the actual solid 

 would assume were the centrifugal force removed, its elasticity 

 being supposed unimpaired and the same as actually exists for 

 smaller distortions. 



4. In order that the principle stated in this form may hold 

 good, it is not necessary that the solid should be symmetrical 

 round the axis of rotation. The couple due to the elastic defor- 

 mation by the centrifugal force can be similarly neglected when 

 the moments of inertia of the solid are all unequal, provided, as 

 above, the oscillation of its axis of rotation remains small, as will 

 be the case if it is spinning round the axis of greatest moment 

 and is but slightly disturbed. Thus, in accordance with Poinsot's 

 theorem, the motion may be represented by the rolling of the 

 modified momental ellipsoid of the solid on a fixed plane ; so that 

 the axis of instantaneous rotation will trace out a small ellipse 

 on the surface of the rotating body and the precession will there- 

 fore be exactly periodic 1 . It follows easily from the Eulerian 

 equations of motion of a solid free from external forcive, that the 

 period of the free precessional motion of the axis of rotation 

 round this ellipse is to that of the rotation of the body in the 



f A'B' )* 



ratio {-tttt — , /N ,„, — -n^r , where A , B , V are the effective mo- 

 \{C -A){C -B)) 



ments of inertia, viz. those that would exist when the strain due 



to the centrifugal force is supposed removed as above; agreeing 



with the result already given where A' and B' are equal. 



5. If when the centrifugal force is thus taken off, the rotating 

 solid became dynamically symmetrical like a homogeneous sphere, 

 there would be no free precession at all : if it became effectively 

 prolate, the precession would be in the negative direction. In 

 the actual case of the Earth, the precession is in the positive 



1 It follows from Poinsot's kinematic representation of the Eulerian motion, 

 that for any solid rotating with no couples acting on it, the motion of the axis of 

 rotation in the body itself is in all cases strictly periodic. It is not difficult to 

 extend this result to the more general case in which there are fly-wheels or other 

 sources of gyrostatic momentum attached to the solid. 



