152-154] KINETIC ENERGY. 243 



by Art. 143 (iv). The first and third terms of this cancel, since at the 

 envelope we have -^ = d^>ldt. Hence for any re-entrant system of vortices 

 enclosed in a fixed vessel, we have 



dP 



(iv), 



with two similar equations. If now the containing vessel be supposed 

 infinitely large, and infinitely distant from the vortices, it follows from the 

 argument of Art. 116 that P is constant. This gives the first of equations (i). 



Conversely from (i), established otherwise, we could infer the constancy of 

 the components P, $, R of the impulse*. 



Rectilinear Vortices. 



154. When the motion is in two dimensions xy we have w = 0, 

 whilst u, v are functions of #, y, only. Hence = 0, TJ = 0, so that 

 the vortex-lines are straight lines parallel to z. The theory then 

 takes a very simple form. 



The formulae (8) of Art. 145 are now replaced by 



d(j> d^lr d(b dty 



u = 7 7 , v = 7 r 7 ( J ), 



dx dy dy dx 



the functions </>, i|r being subject to the equations 



V-fcj) = 0, VfA/r = 2f (2), 



where V x 2 = d*/da? + d*/dy 2 , 



arid to the proper boundary-conditions. 



In the case of an incompressible fluid, to which we will now 

 confine ourselves, we have 



dty dty /QN 



" = -*P " = d (8)> 



where >|r is the stream-function of Art. 59. It is known from the 

 theory of Attractions that the solution of 



V 1 2 i|r=2? (4), 



where f is a given function of a?, y, is 

 _ 1 



7T. 

 * Cf. J. J. Thomson, Motion of Vortex Rings, p. 5. 



162 



