240 VORTEX MOTION. [CHAP. VII 



the elements 88 ultimately vary as R 2 . Hence, ultimately, the 

 right-hand side of (1) vanishes, and we have 



T = const ............................... (3). 



152. We proceed to investigate one or two important kine- 

 matical expressions for T, still confining ourselves, for simplicity, 

 to the case where the fluid (supposed incompressible) extends to 

 infinity, and is at rest there, all the vortices being within a finite 

 distance of the origin. 



The first of these is indicated by the electro-magnetic analogy 

 pointed out in Art. 146. Since = 0, and therefore <l> = 0, we have 



w 2 ) dxdydz 



dH 



by Art. 145 (3). The last member may be replaced by the sum of 

 a surface integral 



pff{F(mw -nv) + G (nu -lw) + H (Iv - mu)} dS, 

 and a volume integral 



du dw\ dv 



At points of the infinitely distant boundary, F, G, H are ultimately 

 of the order R~ 2 , and u, v, w of the order R~ s , so that the surface- 

 integral vanishes, and we have 



Gi,.+ HQ dxdydz .................. (1), 



or, substituting the values of F, G, H from Art. 145 (6), 



T = ** HISIF + 7' + ^' daAyde dx ' dyW - (2)> 



where each volume-integration extends over the whole space 

 occupied by the vortices. 



A slightly different form may be given to this expression as 

 follows. Regarding the vortex-system as made up of filaments, 

 let Ss, Ss' be elements of length of any two filaments, cr, a' 

 the corresponding cross-sections, and CD, co' the corresponding 

 angular velocities. The elements of volume may be taken to be 



