89 91.] INVERSE COMPLEX FUNCTIONS. 93 



A 1 X-L j i c dz . f dx , .di/ 



Again since - is the modulus of -= , i.e. of -yy H- 1 -jr , we 

 q div d(f&amp;gt; d(f&amp;gt; 



have 







HIHI) 



which may, by (48), be put into the equivalent forms 



The last formula, viz. 



I = d(x, y) 



simply expresses the fact that corresponding elementary areas in 

 the planes of z and w are in the ratio of the square of the modulus 



of -7 to unity. Compare Art. 76. 



91. Example 7. Assume 



z c sin w, 



or x = c . ^ sin &amp;lt;f&amp;gt;, 



y c - 9 cos $ 



The curves -\Jr = const, are, Art. 88 (cZ), a system of confocal ellipses, 

 and the curves &amp;lt; = const, a system of confocal hyperbolas ; the 

 common foci of the two systems being the points (+ c, 0). 



Since at the foci we have &amp;lt;f&amp;gt; = J (2n + 1) TT, i|r = 0, n being some 

 integer, we see by (49) that the velocity is infinite there. We 

 must therefore exclude these points from the region to which our 

 formulae apply. If the ellipses be taken as the stream-lines the 

 motion is cyclic ; the circulation in any circuit embracing either 

 focus alone is TT, that in a circuit embracing both is 2?r. 



At an infinite distance from the origin -\Jr is infinite, and the 

 velocity zero. 



When c = this case coincides with that of Example 2. We 



