CURVILINEAR COORDINATES 107 



5.51 Spheroidal Coordinates (Prolate Spheroids): 

 i. z + ip = c cosh (u + iv). 



z = c cosh u cos v., 



2. . , 



{ p = c sinh u sm v. 



The three orthogonal surfaces are the ellipsoids and hyperboloids of revolution, 

 and the planes, 6 : 



cosh 2 u c 2 sinh 2 u 

 3 * - 9 p 2 



c 2 cos 2 v c 2 sin 2 v ~ 



With cos u = X, cos = JJL : 



z = c X u. 



4 



p = c\/(X 2 - i) ( i- M 2 )- 



X 2 - I 2 I - /Z 2 2 I 



5* 1 ~ c 2 (X 2 - M 2 )' ?2 ~ c 2 (X 2 - jit 2 )' 3 ~ c 2 (X 2 - i) (i - 



5.52 Spheroidal Coordinates (Oblate Spheroids): 

 i. p + iz = c cosh(w + iv). 



z = c sinh u sin v. 



2 



p = c cosh M cos v. 

 3. cosh M = X, cos z> = JJL. 



2 , o X 2 - i , 2 



- 



5.53 Parabolic Coordinates: 

 i. z + ^'p = c( + w) 2 . 



z = c(u? v 2 ), 



2 CUV. 

 3. U 2 = \, V 2 = jLt. 



With curvilinear coordinates, X, ju, 0: 



