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 



+ *&amp;gt;) ds , 



............ (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, If, N, respectively, 

 we find, putting 



, dx M , dy . , dz .. 



~J &amp;gt; w TJ = r l w J ~ b &amp;gt; 



ds ds ds 



and replacing the volume-element a Ss by 



P = p$jj(y% zrj) docdydz, L = /o///(2/ 2 + ^ 2 ) f dxdydz, 

 Q = pfff(zg - a) da?dyrf, Jlf = p///(^ 2 + a?) 77 dajrfy^, [-...(12)*, 

 -R = P/J/C^ 7 / ~ ^ f ) docdydz, N = pfff(a? + 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 



_ = ff(lu + mv + nw) pdS .................. (1). 



If the surface S 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 

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



