71 74.] COMPLEX VARIABLES. 71 



For if we introduce a circle of great radius r as the external 

 boundary of the portion of the plane xy considered, we find that 

 the corresponding part of the integral on the right-hand side of 

 (1.0) tends, as r increases, to the value irpM (Mlogr + C), and is 

 therefore ultimately infinite. The only exception is when M = 0, 

 in which case we may suppose the line-integral in (10) to extend 

 over the internal boundary only. 



73. The functions &amp;lt;/&amp;gt; and -^ are connected by the relations 



dx dy* dy dx 



These are the conditions that &amp;lt;-M^, where i stands for V 1&amp;gt; 

 should be a function of the complex variable x + iy. For if 



&amp;lt;l&amp;gt; + i1r=f(x + iy) .......... . .......... (12), 



we have 



~ ......... (13), 



whence, equating separately the real and the imaginary parts, we 

 obtain (11). 



Hence any assumption of the form (12) gives a possible case of 

 irrotational motion. The curves &amp;lt;/&amp;gt; = const, are the curves of equal 

 velocity-potential, and the curves i|r = const, are the stream-lines. 

 Since, by (11), 



dx dx dy dy 



we see that these two systems of curves cut one another at right 

 angles, as we have already proved. See Art. 27. Since the rela 

 tions (11) are unaltered when we write ty for &amp;lt;f&amp;gt;, and &amp;lt;/&amp;gt; for i|r, 

 we may, if we choose, look upon the curves ty = const, as the equi- 

 potential curves, and the curves &amp;lt; = const, as the stream-lines ; 

 so that every assumption of the kind indicated gives us two 

 possible cases of irrotational motion. 



74. The fundamental property of a function of a complex 

 variable, from which all others flow, is that it has a differential 

 coefficient with respect to that variable. If &amp;lt;, ty denote any 

 functions whatever of x and //, then corresponding to every value 



