534 VISCOSITY. [CHAP, xi 



the axis of z being that of rotation. At the surface r a we must have 



u=a>y, v=a>x, w = Q, 

 if o> be the angular velocity of the sphere. This gives A o>a 3 ; cf. Art. 292. 



296. The solutions of the corresponding problems for an 

 ellipsoid can be obtained in terms of the gravitation-potential of 

 the solid, regarded as homogeneous and of unit density. 



The equation of the surface being 



# 2 /2 Z 2 



a* + F+0= l ................................. (0> 



the gravitation-potential is given, at external points, by Dirichlet's formula* 



where A = {( 2 + A)(& 2 + A) (c 2 + A)p ........................ (iii), 



and the lower limit is the positive root of 



This makes 



dQ dQ do. 



(v), 



where 



We will also write 



, rd\ . ... 



X = abcl .............................. (vn); 



it has been shewn in Art. 110 that this satisfies V 2 x = 0- 



If the fluid be streaming past the ellipsoid, regarded as fixed, with the 

 general velocity u in the direction of #, we assume t 



dxdy dy ' 



d 2 Q dy 



w = A -= j + Bx -A 



dxdz dz 



These satisfy the equation of continuity, in virtue of the relations 



* Crelle, t. xxxii. (1846) ; see also Kirchhoff, Mechanik, c. xviii., and Thomson 

 and Tait, Natural Philosophy (2nd ed.), Art. 494 w. 

 t Oberbeck, /. c. ante p. 529. 



