Let the ellipsoid rotate about the a;- axis with angular velocity co . Then, by substitut- 

 ■ Jl, m, n in Equation [135d], and also u- 

 the boundary condition to be satisfied by is 



ing for i, m, n in Equation [135d], and also u- -d<f>/dx, v= -d<f>/dy, w= -dcf)/dz, co =cu =0, 



X d(f) y d(j} z d<f> / 1 1 



fl2 d^ b^ dy ^2 d2 """*'- - 1^^- 



A solution of the last equation which is also a solution of the Laplace Equation or 

 Equation [7a] is 



b^-c^ 



b^ + c' 



CO vs. [140b] 



The components of velocity are 



b^-c^ b^-c^ 

 w = 0, V = CO z, w CO y. [140c, d, e] 



The flow thus proceeds in planes perpendicular to the a^-axis, and it is the same in all of 

 these planes, except for variation in the size of the occupied elliptical cross section. The 

 flow pattern is, in fact, the same as that inside an elliptical cylinder rotating about its axis, 

 as illustrated in Figure 173. 



The kinetic energy of the fluid is 



pcj^ I 1 2 ^2X2 rrr o„ /a2 „2\i 



7' = /// {y^+z^) dxdydz = — p abc —co^. [140f] 



2 \ b^ + c^J JJJ 15 i,z^^. 



To evaluate the integral, substitute x = ax', y = br 'cos d, z = cr'sin 6. 



Analogous results hold for rotation about the y- or 2-axes; and by combining rotations 

 about the coordinate axes the general case can be represented of rotation about any axis 

 passing through the center of the ellipsoid. 



The axes rotate, of course, with the shell. 



(See Reference 1, Article 110; Zahm.^^'*) 



372 



