112] ROTATION OF AN ELLIPSOID. 165 



The formulae for the cases of rotation about y or z can be written 

 down from symmetry*. 



The formula for the kinetic energy is 



fdS 

 dn 



= pCp . I - i - - o . I i(ny mz) yzdS, 

 Jo (a 2 + X)*(6 2 + X)*(c 2 + X)* lr 



if (I, m, n) denote the direction-cosines of the normal to the 

 ellipsoid. The latter integral 



= fff(f - z 2 ) dxdydz = (6 2 - c 2 ) . 

 Hence we find 



The two remaining types of ellipsoidal harmonic of the second order, finite 

 at the origin, are given by the expression 



where 6 is either root of 



1 1 1 



this being the condition that (i) should satisfy v 2 $ = 0. 



The method of obtaining the corresponding solutions for the external 

 space is explained in the treatise of Ferrers. These solutions would enable us 

 to express the motion produced in a surrounding liquid by variations in the 

 lengths of the axes of an ellipsoid, subject to the condition of no variation of 

 volume 



d/a+b/b + c/c = (iii). 



We have already found, in Art. 110, the solution for the case where the 

 ellipsoid expands (or contracts) remaining similar to itself ; so that by super- 

 position we could obtain the case of an internal boundary changing its 

 position and dimensions in any manner whatever, subject only to the con- 

 dition of remaining ellipsoidal. This extension of the results arrived at 

 by Green and Clebsch was first treated, though in a different manner from 

 that here indicated, by Bjerknest. 



* The solution contained in (5) and (6) is due to Clebsch, "Ueber die Bewegung 

 eines Ellipsoides in einer tropfbaren Fliissigkeit," Crelle, it. Iii., liii. (1856 7). 

 t 1. c. ante p. 156. 



