The inverse transformation to Equation [81a] is 



1 i/j 



3"= — [z±(2^ -ic^) ]; [81q] 



the plus sign goes with r>c or ^>6, and the negative with r<c or ^<d, as is easily verified for 



positive real s; for a real z<- 2 c, however, the symbol (2 - 4 c^) must be understood to stand 

 for the negative square root. 



(See Reference 2, Section 6.30, where c is replaced by c/2, also Section 6.32.) 



82. ELLIPTIC COORDINATES 

 Let 



2 = a; + zy = c cosh ^, 1^ = ^ + z 77. [82a, b] 



This transformation was studied briefly in a different notation in Section 61, and the results 

 obtained there will be assumed. From Equations [61b, c, g, h, i, j] 



X = c cosh ^ cos -q, y ^ c sinh f sin rj, [82c, d] 



cosh f = t[(a;+ c)2 + /] + [(3. _ c)2 + y'^]^'\, [82e] 



2c 



cos 7; = — t[(a; + c)2 + y^]^' - Ux - cf + yH^'\, [82f] 



2c 



a;^ ip- x^ V^ 

 + = 1, =1. [82g,h] 



c^ cosh-^ ^ c^ sinh^ ^ c^ cos'^ 7/ c-^ sin^ 77 



If ^ is held constant while r\ is given all possible values, an ellipse is obtained on 



the 2-plane, with semimajor and semiminor axes 



\ 



a' ^ c cosh ^, 6'= c |sinh ^. [82i, j] 



The same ellipse is obtained for f = - ^j as for ^= ^j. If 7? is held constant while granges 

 from -00 to 00, a hyperbola is obtained with semiaxes 



a"- c |cos r]\, h" = c |sin 77]. [82k, 1] 



All ellipses and hyperbolas have common foci at {- c, 0), and 



a'2 _ 6'2 ^ c^, a"2 + 6"2 = c^. [82m, n] 



192 



