162 VORTEX MOTION. [CHAP, VI. 



if I, m, n be the direction cosines of the inwardly-directed normal 

 to any element dS of the boundary. The part of this expression 

 which contains p gives the rate at which the external pressure 

 works, the remaining part expresses the rate at which the mass is 

 losing potential energy. If the mass be enclosed within fixed rigid 

 walls, we have 



lu + mv + nw = 



dT 



at the boundary, and therefore -j- = 0, or T= const. The same 



Cut 



result holds for the case of an unlimited mass of liquid subject to 

 the conditions of Art. 129. We then have, beyond the vortices 



p + pV=-p-~ 



and it appears from Chapter ill. that at an infinite distance from 



the origin &amp;lt;j&amp;gt;, and therefore also -~ is constant with respect to 



out 



x, y, z. Under these circumstances the surface integral in (32) is 

 zero. Compare Art. 65. 



138. We proceed to apply the foregoing general theory to the 

 discussion of some simple cases. 



1. Rectilinear Vortices. 



Suppose that we have an infinite mass of liquid in motion in 

 two dimensions (xy), so that -u t v are functions of x, y only, and 

 w = 0. We have then f = 0, 77 = everywhere and therefore also 

 L = 0, If = 0. The value of N is 



*=J- 



and if we perform the integration with respect to z between the 

 limits + 7, and then make 7 infinite, we find 



N = -L log *y // S dxdy - // ? log r dx dy, 



where r now (and as far as Art. 140) stands for {(x x) z + (y y) 2 }*. 

 The first term in the value of N, though infinite, is constant, and 



