150-151] COMPONENTS OF IMPULSE. 239 



For the similar transformation of (9) we must have recourse 

 to Stokes' Theorem ; we obtain without difficulty the forms 



(11). 



From (10) and (11) we can derive by superposition the com- 

 ponents of the force- and couple- resultants of any finite system of 

 vortices. Denoting these by P, Q, R, and Z, M, N, respectively, 

 we find, putting 



m a a-', 



,dx ,dy , ,dz' 



and replacing the volume-element o-'Ss' by 



- *?) dxdydz, L = pfff(f + * 2 ) f dxdydz, 

 -xQ dxdydz, M =pfff(z*+ a?) r) dxdydz, [...(12)*, 

 R = pfff(xr) y%) dxdydz, N = /"JJ(# 2 + 2/ 2 ) f dxdydz 

 where the accents have been dropped, as no longer necessary. 



151. Let us next consider the energy of the vortex-system. 

 It is easily proved that under the circumstances presupposed, and 

 in the absence of extraneous forces, this energy will be constant. 

 For if T be the energy of the fluid bounded by any closed surface 

 S, we have, putting F = in Art. 11 (5), 



DT 

 ~- = jj(lu + mv + nw) pdS .................. (1). 



If the surface 8 enclose all the vortices, we may put 



and it easily follows from Art. 148 (4) that at a great distance R 

 from the vortices p will be finite, and lu + mv+nw of the order 

 R~ 3 , whilst when the surface 8 is taken wholly at infinity, 



* These expressions were given by J. J. Thomson, On the Motion of Vortex 

 Eincfs (Adams Prize Essay), London, 1883, pp. 5, 6. 



