137. OVARY ELLIPSOIDS (OR PROLATE SPHEROIDS) 



Problems involving an ellipsoid are most easily handled in terms of some form of ellip- 

 soidal coordinates. Special types are used when two axes are equal. 



For an ovary ellipsoid, or prolate spheroid, the prolate-spheroidal coordinates i^, n, a 

 are most conveniently defined inversely, thus: 



X = kfii^, y = gTcos oj, 2 = oT sin w, [137a, b, c] 



o 1/2 T 1/2 



S'=^(<C^-1) (l-/i^) , [137d] 



where k is an arbitrary positive constant and positive values of the radicals are intended. Thus 



r^ = a;2 + y- + 3^ = a;^ +~2 = k^ {C^ + t^^ - !)• [137e] 



Here o) is an angle representing position about the a;-axis; (,, y., and &j are dimensionless, 

 whereas k represents a fundamental length. The coordinate surfaces for ^and ^l are confocal 

 ellipsoids and hyperboloids of revolution, with foci on the a;-axis 0.1 x = - k\ their equations, 

 obtained by eliminating either // or C,i ^■'■e 



.2 ~2 ^2 -^2 



=1. [137f,gJ 



^2^2 ^2^^2_^^ ^2^2 ^2^^_^2^ 



The traces of the coordinate surfaces on any plane through the a;-axis are confocal el- 

 lipses and hyperbolas; it was seen in Section 61 that such curves cut each other orthogonally. 

 It will be simplest to treat the two halves of such a plane as separate planes, distinguished 

 by complementary values of co. On each half-plane either x and oTor ^ and p. then serve as 

 single-valued coordinates and ar> 0. Convenient ranges of values for ^ and ft, as indicated 

 in Figure 228, are: 1 ^ t^, - 1 < // < 1. 



The coordinates 4 f- ^-^e simply the elliptic coordinates of Section 82 in disguise: 

 (^ = cosh ^, /i = cos 77, and here k = c. Formulas for t^and /i in terms of x and y can be written 

 down at once from Equations [82e,f]. 



The semiaxes of any <^ ellipsoid and its ellipticity are 



a'=kC, b' =k{C^ -I) ^^ , e'^l/C [137h,i,j] 



In terms of these, x = a'y,7o ^ b' -Jl - [i . Also, k = a' e '. On the a;-axis, for a; ^ A, fi = 1, 

 x = k(l;^ for x^-k, /:/ = -l, x = - k C- For |a;| < k, ^ = 1, a' = k and x = k\i. On the oT-axis, 

 // = and ar= /c \/4^ - 1 • 



346 



