44 ME. J. H. JEANS ON THE POTENTIAL OF ELLIPSOIDAL BODIES, AND 



A physical interpretation of these first few solutions can readily be found. For 

 the undisturbed ellipsoid of axes ka, kb, kc, and origin at x , y , z , 



, - 



~ ~ 



C 



and the special ellipsoid which has been under consideration has been that for 

 which x y = z = 0, k = 1. We can change the centre and axes of the ellipsoid 

 contemplated in equation (55) by varying x a , y , z , a 2 , l> 2 , c 2 , and k. If we change 



k 2 by an amount 3k 2 in equation (55), the change in $ is given by M> = 



so that f may be regarded as replaced by f+<f> where </> = Sk~. Thus the solution 

 n = represents a change from k 2 to IC*K ; physically it represents a change in 

 the scale of the ellipsoid. 



Similarly, if in equation (55) x u is changed by <fcr () , y {l by fy , and z by Sz a , we find 



that SQ = -2y, (A) (^ + ( ^f + ^\ so that 



A B ( 



and the solution n = 1 is seen to represent a motion of the centre of the ellipsoid. 

 Similarly, if we put ,r u = y n = z = and k = 1 in equation (55) so that <J> = 

 and differentiate logarithmically with respect to a 2 , we obtain 



_ _ 



V/Sa 2 " 2 a 2 2 A 



whence 



Clearly, then, the first three terms in the solution n = 2 represent a distortion of 

 the original ellipsoid produced by a change in the lengths of the axes, and it is 

 easily seen that the complete solution represents a change of this kind combined 

 with a small rotation of the axes. 



n = 3. There are ten terms in the general cubic function of f, y, f. For the 

 present purpose it is convenient to regard this general cubic function as made up 

 of a term e^f, and the sum of three expressions such as 



For the solution given by % = e fr, we have V 2 ^ Wj = 0, so that v l = and 



