318-321] 



Conjugate Functions 



265 



and the transformation is seen to be 





agreeing with that which has already been seen in 317 to give a parabola 

 as a possible equipotential. 



321. As a second example of this method, let us consider the ellipse 



The coordinates of a point on the ellipse may be expressed in the form 



x a cos <f>, y = b sin <, 

 and the transformation is seen to be 



z = a cos W + ib sin W. 



FIG. 89. 



We can take a = c cosh a, b = c sinh a, where c 2 = a 2 - 6 2 , and the trans- 

 formation becomes 



z c cos ( W + ia} = c cos { U+ i (V + a)}. 

 The same transformation may be expressed in the better known form 



z = c cosh W. 

 The equipotentials are the confocal ellipses 



tf+x^+x*' 1 ' 



