1896.] 



The Rotation of an Elastic Spheroid. 



187 



3. To obtain a solution of the equations of motion subject to these 

 boundary conditions, suppose the displacements u, v, iv consist of two 

 parts : 1st, u . v , Wo, due to the rotation of the body as a whole 

 through, small angles 6 U 2 about Ox, Oy ; and 2nd, u Xi i\, Wi, due to 

 elastic distortion. Then — 



n — m +¥i = ^2 + «i> 



w = w + Wi = xO z — yOi+W\. 



Changing the variables from u, v, to to u h Vi, W\ the equations of 

 motion take the form 



(«V 2 + ^)^i-2"-'^Mi 



ax ay dz 



dx 



—\ 2 ze>-\-2ivi\ze 1 



3^ 



\Z0, 



+ \-(xe i —ij6 1 ) 



while the boundary conditions become 



■^cosa + ' 



/ dih 



diii 



dui 



diii 



dvi 



cos ft + — : cos 7 + — cos a + —cos ft 



dx 



+ cos 7 | — OA — 9fsi) cos a = oj-(0 2 xz—6iyz) cos a: 



, _ . f cfoi , cZt'i , , : d»i , du x , d^i 



y v os /3 + »& < — cos a + — cos # H — — cos 7 + — cos a + — 7 — cos ft 

 I dx dy ' dz dy dy 



dw x 



dw 1 I . 1 



+ — cos 7 j> — (v 1— gr£V) cos /3 = w\0 2 xz — 0\yz) cos /3 



>>(*), 



\> cos 7 + ™ :: T - ! - cos a + ~- cos j8 + cos 7 + ^ cos oc + cos /3 

 [ aa; dy dz dz dz 



dwi 



dv x 



f dw x 



dx 



l —— cos 7 > — (Vi — </<Ti) cos 7 = ic-(0 2 xz—6 l yz') cos 7 



Ct2 J 



+ 



where & denotes the height of the surface waves consequent on the 

 displacements u u r 1? te ls and v\ the part of v due to the harmonic 

 inequalities fa. The advantage of this transformation consists in the 



2 



