99-100] ELLIPSOIDAL HARMONICS. 145 



The dynamical consequences of the formula (vii) will be considered more 

 fully in Art. 140 ; but in the meantime we may note that if the sphere be 

 held at rest, so that u, v, w = 0, it experiences a force tending to diminish the 

 energy of the system, and therefore urging it in the direction in which the 

 square of the (undisturbed) fluid velocity, u 2 + v Q 2 +w Q 2 , most rapidly increases*. 

 Hence, by Art. 38, the sphere, if left to itself, cannot be in stable equilibrium 

 at any point in the interior of the fluid mass. 



Ellipsoidal Harmonics. 



100. The method of Spherical Harmonics can also be adapted 

 to the solution of the equation 



V 2 = .............................. (1), 



under boundary-conditions having relation to ellipsoids of revo- 

 lution *f". 



Beginning with the case where the ellipsoids are prolate, we 



write 



x = k cos 6 cosh i) k^ } 



y = CT cos &>, = 37 sin a>, I ......... (2). 



where tsr = k sin sinh 77 = k (I - /* 8 )* (f 2 - 1)* j 



The surfaces f = const., //. = const., are confocal ellipsoids, and 

 hyperboloids of two sheets, respectively, the common foci being the 

 points ( k, 0, 0). The value of f may range from 1 to oo , whilst 

 /A lies between 1. The coordinates //-, f, &> form an orthogonal 

 system, and the values of the linear elements $ M , Bs^ Bs^ described 

 by the point (x, y, z) when //,, f, co separately vary, are respectively, 



^)*(?-I)*Ba> .................. (3). 



To express (1) in terms of our new variables we equate to zero 

 the total flux across the walls of a volume element 

 and obtain 



d d6 ~ d d< d d6 . 



* Sir W. Thomson, " On the Motion of Kigid Solids in a Liquid &c.," Phil. 

 Mag., May, 1873. 



t Heine, "Ueber einige Aufgaben, welche auf partielle Differentialgleichungen 

 fiihren," Crelle, t. xxvi., p. 185 (1843); Kugelfunktionen, t. ii., Art. 38. See also 

 Ferrers, Spherical Harmonics, c. vi. 



L. 10 



