whence 



k = e' c' = ^Jc''^~a''^, C= a'/k ^ a'/ yjc''^ -a'^ = ^/l-e'^/e\ [138j, k] 



a '/i, w = c'{l - 11 ) 



2x1/2 



[1381, m] 



The ellipsoid for (;^ = is a circular disk of radius tff = c ' = kin the y2-plane, on which 

 ST = A; V 1 -|L(^; the remainder of the y2-plane is the hyperboloid /^ = 0, on which TS = k \j l^ + 1. 

 The hyperboloid for /:i = - 1 is the entire ^--axis, on which x -- kC,. 



Toward infinity, ^-» oo and, approximately, 7S = ^^(1 -/i^)^'^^, r = {x^ +W^)^^-^ = A;^, so 

 that ^ = r/A;, \i = x/k^ = a?/?' = cos 6 in terms of the polar angle 0. 



The coordinate elements of distance are 



C^ + 1 



5s/-=/t|-! —I 5^, Ss =a' '^ 



2 _^ , 2\ 1/2 



1-f.' 



Sfi, 



[138n, o] 



5s, 



r5co = /fc(4^ + l)i/^ (1-^^)1/^ S 



[138p] 



The coordinate directions for (^ and ^ make angles d^, with the positive a;-axis which lie 

 in the ranges O-df-^n, -— =6 = — and are given by 



2^2 



cos V = -sin 6 = pi 



^^ + 1 



<= +fi 



[138q] 



and the velocity components in the coordinate directions are given by Equations Ll37o, p, q] 

 or 



doj dcf) 





V 



[138r, s, t] 



The Laplace equation, and the relations between <;i and i// if i// exists, are 





(C^ + i)— ^ 



d 





'^mJ (^2^i)(i_^,2) ^^^2 



= 0, [138u] 



