108 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 



7/ I 



4 . 



5.54 Toroidal Coordinates: 



, z + a + ip 



i. + w = log - A 



& z - a + ip 



a sinh w 



P = 



2. 



cosh w cos v 

 a sin v 



, 

 hi 



cosh u cos v 

 cosh u cos v cosh u cos 



a a smh w 



The three orthogonal surfaces are: 



(a) Anchor rings, whose axial circles have radii, 



a coth u, 



and whose cross-sections are circles of radii, 



a csch u\ 



(b) Spheres, whose centers are on the axis of revolution at distances, 



a cot w, 

 from the origin, whose radii are, 



a esc v, 



and which accordingly have a common circle, 



p = a , z = o; 



(c) Planes through the axis, 



w = 6 = const. 



