160] 



CONFORMAL REPRESENTATION. 



319 



We see that the straight lines x = const, correspond to con- 

 focal ellipses, in (I) and (II), while the lines y = const, correspond 

 to confocal hyperbolas, while in (III) and (IV), the lines 

 y = const, correspond to ellipses and as = const, to hyperbolas. 

 The geometrical character of all four transformations being 

 therefore the same, we shall consider only case I, Fig. 66. 

 (It is to be noticed that we have interchanged z and w in (13).) 

 To any line x const, corresponds the ellipse whose semi-axes 

 are cosh x and sinh x. When x the ellipse reduces to the 



FIG. 66. 



straight line between the focal points u=I, v = and u = l, 

 v = 0. As x increases the ellipses become continually larger, until 

 for x infinite they become infinite circles. For any negative x we 

 get the same ellipse as for the corresponding positive. The lines 

 y = const, correspond to the hyperbolas whose semi-axes are cos y 

 and sin y. When y = the hyperbola reduces to those portions of 



