Hie Rotation of an Elastic Spheroid. 



185 



January 16, 1896. 



Sir JOSEPH LISTER, Bart., President, in the Chair. 



A List of the Presents received was laid on the table, and thanks 

 ordered for them. 



The following Papers were read : — 



I. " The Rotation of an Elastic Spheroid." By S. S. HOUGH, 

 B.A., Fellow of St. John's College, and Isaac Newton 

 Student in the University of Cambridge. Communicated 

 by Professor G. H. Darwin, F.R.S. Received December 

 12, 1895. 



(Abstract.) 



1. It is well known that if a rigid body, whose principal moments 

 of inertia are A, A, C, be set rotating about its axis of symmetry and 

 then be subjected to a slight disturbance, it will execute oscillations 

 about its mean position, in consequence of which the axis of rotation 

 will undergo periodic displacements relatively to the body in a period 

 which bears to the period of rotation the ratio A : C— A. The object 

 of the present investigation is to determine to what extent this 

 period will be modified if the body, instead of being perfectly rigid, 

 is capable of elastic deformations. 



The analysis is confined to the case of a homogeneous spheroid of 

 revolution composed of isotropic, incompressible, gravitating material, 

 while for further simplicity the body is supposed to be free from 

 strain in its interior when rotating uniformly. If we refer to a set of 

 rectangular axes rotating with uniform angular velocity w about the 

 axis of z, the rigorous differential equations f or the vibrations of such 

 a body are 



dyjr ~) 

 + ; — = u — 2ivv I 

 ox j 



, u ! 



% I a), 



2 , - 



n^j~w -f ~ = w 

 Oz 



du . dv dw 



— H — rr = 



ax dy az 



y 



VOL IIX. 



