160] 

 which give 



Vo, 2 + //, = cos v . Va 2 b 2 , 



CONFORMAL REPRESENTATION. 



317 



sin v . \/6 2 a 2 . 



Inserting the values of the four square roots in the values of x 

 and y in (2) 



x = {e w + (a 2 - 6 2 ) e~ M } cos v, 

 y = i {e u - (a* - 6 2 ) e~ u ] sin v, 



(12) 3? + M/ = i [e M (cos i; + i sin t>) + (a 2 - 6 2 ) e~ w (cos v i sin v)] 



= i. [ e +*> + ( a 2 _ # 



which gives the form of the function sought, 



(13) * = i{^+(a 2 -6 2 )e- w }, 

 or 



(14) w = log {z *!& - (a 2 - ft 2 )}. 



We may now conveniently change our unit so that the focal 

 distance Va 2 b 2 shall equal unity. Then the function z becomes 

 the hyperbolic cosine of w. A table of comparison of the principal 

 properties of the hyperbolic and circular functions is appended*. 



* 160 A. Hyperbolic and Circular Functions. 



