

L/2 /A2^.a/2 /,2^^x3/2 



^0 (a^ + A)'^^ (J^ + A)^/^ (c^ + A)-^ 



If one axis is reduced to zero, the ellipsoid becomes an elliptical disk. 



(See Reference 1, Articles 112-115; Durand,^ Volume I, p. 293; Tuckerman;^^^ Zahm.^""*) 



142. ELLIPSOID CHANGING SHAPE 



If the semiaxes of the ellipsoid defined by the equation 



2 2 2 



X"^ y^ 3^ 



— + — + 1=0 [142a] 



a^ b^ c2 



change with time at the rates a, b, c, without rotation of the ellipsoid and with its center at 

 rest, and if a point moves with components of velocity x, y, z, in such a way as to remain al- 

 ways on the surface of the ellipsoid, then at this point Equation [142a] is always satisfied, 

 and, differentiating Equation [142a] with respect to the time, 



2 2 2 



X . y . z . x^ . y • z . 



— x + — y + z - — a b - — c = 0. [142b] 



a2 62 ^2 ^3 j3 ^3 



Now according to Equations [135f, g, h] the direction cosines of the normal to the ellipsoid 

 at the point x, y, z are 



X y z 



I = 2k — , m - Ik — , n = 2k — , 



a} b^ c2 



where A; is a constant of proportionality. After substituting in Equation [I42b], the combina- 

 tion Ix + my + nz occurs; this represents the normal component of the velocity of the surface, 

 which must equal the same component of the fluid velocity or 



lu + mv + nw = - \ I — + m — + n — 

 dx dy dz 



m 



