nc nc r , -, 



x' - sinh nk, y - sin na, L88h,i] 



2g' 2g' 



2 TlA . 2 ^f^ •'■ 



g' = sinh + sin = — (cosh nk- cos nu), [88i] 



^ 2 2 2 



from hyperbolic formulas listed in Section 32. Similarly 



z = c coth — , [88k] 



2 



c c 1 



X = — sinh k, y - — sin ii, g = — (cosh k - cos /x). L881, m, nj 



2g 2g 2 



Thus fi has the sign of y and - n- ^ pi < it. The variables A, /x are sometimes called bipolar 

 coordinates on the 2-plane. 



The transformation can be visualized by imagining the s-plane to be cut along the real 

 axis between - c and to be pulled or pushed until all arcs come into the proper position, with 

 the remainder of the real axis retaining its direction. 



An important special case is that of a circle through {- c, 0) on the s-plane, such as 

 AB, which transforms into two arcs meeting at an interior angle 277 - S|y1 or (2 - n) n. If 

 n = 2, these coalesce into a single arc, as in Section 78. If < n < 2, the exterior of the 

 circle is mapped conformally onto the part of the 2 '-plane lying outside of the crescent 

 enclosed by the two arcs. If < ti < 1, the ends of the "crescent" are reentrant; compare 

 Figure 142b. 



Toward infinity. Equation [88a] becomes, by binomial expansion, 



(-^••■•)/(-f--)=(-f--)/(-f--)- 



Hence at infinity z' -* z and the two planes agree. 



The transformation fails to be conformal, in general, at s = - c. From Equations 

 [88g] and [88k] 



dz' dz' ( dz\~'^ 1 . -,1 C / . ,1 nC 

 = n sinh — / sinh 



dz dC \dC I 2 / 2 



212 



