in terms of the velocity potential (j>. Hence, after eliminating /, m, n from the last expression, 

 Equation [142b] gives for the boundary condition for cf> 



1 ^ +A^ + ±^ +—a +^'b + — ^ = 0. [142c] 



a2 dx j2 dy ^2 dz ^3 j3 ^3 



This equation is satisfied by 



<h^ - -\—x^ +-y^ + —2^ ] , [142d] 



^ 2 la h^ c ' 



which is a solution of the Laplace Equation or Equation [7a] provided 



a b c 

 — + — + — = 0. 

 a h c 



But this is merely the condition that the volume of the fluid enclosed in the ellipsoid or 

 4/7 ahc/?> shall remain constant, so that (d/dt) [log^ (abc)] = 0. 

 The velocity components of the fluid are 



d(j) a b c 



M = - = — X, V = -^-Vj ^ - — 3. [142e, f, gj 



dx a b c 



(See Reference 1, Article 110.) -. 



143. FLOW PAST A PARABOLOID 



Consider the steady flow parallel to the axis of a solid body having the form of a pa- 

 raboloidal solid of revolution. With the origin at its focus and the a;-axis of cylindrical coor- 

 dinates along its axis, let the equation of the surface of the solid be 



w^ = a2 _ 2ax, [I43a] 



where IS denotes distance from the axis. Its apex is at a; = a/2 and it extends toward nega- 

 tive x. Let the fluid approach at velocity (J from a; = +<». 



375 



