534 VISCOSITY. [CHAP, xi 



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



u = ay, v = ax, w = Q, 

 if &amp;lt; be the angular velocity of the sphere. This gives A = &amp;lt;oa 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 



5+1 



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





where A = {( 



and the lower limit is the positive root of 



This makes 



dQ dQ, dQ. 



-j = zirax, -y- = 27raV, r~ 

 dx ciy ciz 



where 



d\ 



We will also write 



. ... 



.............................. (vn) ; 



it has been shewn in Art. 110 that this satisfies V 2 x~^- 



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

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



-, ..................... (viii). 



dxdy dy ( 



-,- 

 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. 494m. 

 t Oberbeck, I.e. ante p. 529. 



