J56 PROBLEMS IN THREE DIMENSIONS. [CHAP. V 



ponents are u, v, w and if p, q, r be the angular velocities about 

 the principal axes, we have by superposition 



6 2 - c 2 c 2 - a? a 2 - b 2 



d&amp;gt; = ux vy vrz r , -- r p yz - -- - qzx -- r rxii 

 6 2 + c 2 c 2 + a 2 ^ a 2 + 6 2 &quot; 



........................ (3)* 



We may also include the case where the envelope is changing 

 its form as well as position, but so as to remain ellipsoidal. If the 

 axes are changing at the rates a, b, c, respectively, the general 

 boundary condition, Art. 10 (3), becomes 



a + + ,c + + + = 



a 3 b 3 c 3 a 2 dx b 2 dy c 2 dz 



which is satisfied by 



The equation (1) requires that 



a b c 



which is in fact the condition which must be satisfied by the 

 changing ellipsoidal surface in order that the enclosed volume 

 (^jrabc) may be constant. 



108. The solutions of the corresponding problems for an 

 infinite mass of fluid bounded internally by an ellipsoid involve 

 the use of a special sj^stem of orthogonal curvilinear coordinates. 



If x, y, z be functions of three parameters X, /A, v, such that the 

 surfaces 



X = const., /ji = const., v = const (1) 



are mutually orthogonal at their intersections, and if we write 



* This result appears to have been obtained independently by Beltrami, 

 Bjerknes, and Maxwell, in 1873. See Hicks, &quot;Report on Recent Progress in 

 Hydrodynamics,&quot; Brit. Ass. Rep., 1882. 



t Bjerknes, &quot; Verallgemeinerung des Problems von den Bewegungen, welche in 

 einer ruhenden unelastischen Fliissigkeit die Bewegung eines Ellipsoids hervor- 

 bringt,&quot; Gottinger Nachrichten, 1873. 



