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 I 7 of a portion of fluid bounded by a 

 cylindrical surface whose generating lines are parallel to the axis 

 of z t 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 



d<f)/dn = d-fr/ds, 

 the formula (1) may be written 



tT=pfW (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 (2irJ/") 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 

 npM(Mlog r- C\ and is therefore ultimately infinite. The only exception is 

 when Jf=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 </> and ty are connected by the relations 



d$ _dty d<j)__(ty ,-. 



doc dy ' dy dx ' ' 



These are the conditions that <j) + ity, where i stands for \f 1, 

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



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

 mische und Neivton'sche Potential, Leipzig, 1877. 



