60-62] KINETIC ENERGY. 73 



The reader who is interested in the matter will have no difficulty in working 

 out a theory of two-dimensional sources and sinks, similar to that of Arts. 

 5658 *. 



61. The kinetic energy T of a portion of fluid bounded by a 

 cylindrical surface whose generating lines are parallel to the axis 

 of z y and by two planes perpendicular to the axis of z at unit dis 

 tance apart, is given by the formula 



where the surface-integral is taken over the portion of the 

 plane xy cut off by the cylindrical surface, and the line-integral 

 round the boundary of this portion. Since 



the formula (1) may be written 



2!T=p#ety ........................ (2), 



the integration being carried in the positive direction round the 

 boundary. 



If we attempt by a process similar to that of Art. 46 to calculate the 

 energy in the case where the region extends to infinity, we find that its value 

 is infinite, except when the total flux outwards (2rJO is zero. 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 (I) tends, as r increases, to the value 

 rrpM(Mlogr- C\ and is therefore ultimately infinite. The only exception is 

 when M = 0, in which case we may suppose the line-integral in (1) to extend 

 over the internal boundary only. 



If the cylindrical part of the boundary consist of two or more 

 separate portions one of which embraces all the rest, the enclosed 

 region is multiply-connected, and the equation (1) needs a correc 

 tion, which may be applied exactly as in Art. 55. 



62. The functions &amp;lt; and ^ are connected by the relations 



d(f&amp;gt; _ dty d(f) _ d^r , . 



doc dy dy dx 



These are the conditions that (fr + iifr, where i stands for V~l&amp;gt; 

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



* This subject has been treated very fully by C. Neumann, Ueber das logarith- 

 misclie und Newton sche Potential, Leipzig, 1877. 



