The surfaces ^= constant are again ellipsoids of revolution, but those for ji = constant are 

 now hyperboloids of one sheet, with circular apertures lying in the ys-plane; the equations 

 are 



;2/2 7.2//-2 



k^C k^iC +1) 



^2(1-^2) 



[138e, f] 



In any plane through the a;-axis the intercepts are orthogonal ellipses and hyperbolas with 

 common foci lying on the focal ring defined hy x = 0, cj = k. See Figure 232, on which again 

 only a half-plane is shown. Assume ^ = 0. Then - 1 = fx = 1. 



Figure 232 - Choice of signs for oblate-spheroidal coordinates. 

 See Section 138. 



The relation with the elliptic coordinates of Section 82 is now: k = c, i^ = sinh ^, 

 fi = sin T]-, and a- and y are replaced, respectively by oTand x. Formulas for <; and fi in terms 

 of X and y are easily written down from Equations [82e, f]. 



The polar and equatorial radii of any ellipsoid, in the x and To directions, respectively, 

 and the eccentricity of its meridian section are 



a'=kC c'=k{e + l)'/\ e'= ^c'^-a'^/c'= l/ie + l)'^\ [138g, h, i] 



357 



