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 



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

 ellipsoid. The latter integral 



= fff(y 2 - z *) dxdydz = (6 8 - c 2 ) . f Tra&c. 

 Hence we find 



S P P &quot; 



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 



111 



(ii), 



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 



= .............................. (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, &quot;Ueber die Bewegung 

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

 t 1. c. ante p. 156. 



