0.2 y2 ^2 



+ =1, a = T + — , 6 = 



„2 a2 r 



c2 

 r - 



The equation of the transformed curve can also be written 



[81h,i,i] 



which shows that the given 2 "-circle becomes an ellipse on the 3-plane having semiaxes c, 

 b. Its foci are located at y = and x = - \/ a - b -- 2c. 



Similarly, a radius from the origin of z'\ on which is constant, becomes a curve on 

 which, from Equation [81f, g] 



y 2., ^ r^ = — , —^ '- = 1. [81k] 



cos d sin 6 cos d sin 6 r ^^2 ^^^2 q ^^2 ^^^2 q 



This represents a hyperbola having semiaxes 2 c cos 0, 2 c sin 6, and foci likewise at 



(i 2 c, 0). Since the transformation is conformal, the hyperbolas and ellipses are orthogonal, 



as were the original circles and radii. 



Toward infinity, 2 -» 2 "and the two planes become alike. The ellipses then reduce 

 to circles like those on the 2 "-plane, and the hyperbolas approach their asymptotes, which 

 have the directions of the original 2 "radii. 



In working with these curves, it is convenient to change somewhat the variables that 

 characterize them. Let a new complex variable ^ be defined by 



z" = ce^, C= ^ + ir]- [811, m] 



Since ^= In {z"/c) and 2"= re , 



^=\n{r/c),r, = e. [81n,o] 



Substitution in Equation [81a] then gives 



2 = 2 c cosh C- [81p] 



This latter transformation will be studied in the next section. 



Each of the ellipses previously described now corresponds to a certain numerical 

 value of ^. There are two different circles on the 2 "-plane corresponding to each ellipse, 

 however, one lying inside of the circle r = c and the other outside of it; their radii Tj, t^ are 

 such that r, r = c2, since in Equation [81i,j] two such values of r give the same a and 6, 

 and Equation [81n] shows that for the larger circle ^> while for the smaller ^ < 0. Each 

 hyperbola corresponds Xjo -q ^ -q - 2nn where 77 j is a real number and n is an integer or zero. 



191 



