160 VORTEX MOTION. [CHAP. VI. 



This triple integral may, exactly as in Art. 129 (12), be replaced 

 by the sum of a surface-integral 



pff{L (nv - mw) 4- M (ho - nu) + N (mu -lv)}dS ...... (26), 



and a volume-integral 



[f[( T (dw dv\ -nrfdu dw\ , T fdv du\] 7 7 7 

 p \\\ \L -= -- -y- + M [-J -- T- + N N -- T ~ H dxdydz 

 JJJ ( \dy dz) \dz dxj \dx dy}} 



......... (27). 



Now it appears from (11) that at an infinite distance R from the 

 origin, L, M, N are at most * of the order -^ , and therefore u, v, w 



.at most of the order -=^ , whereas when the external bounding sur- 

 _tt 



face is increased in all its dimensions without limit the surface- 

 elements dS increase proportionately to R* only. The surface- 



integral (26) is therefore of an order not higher than ^ , and 



therefore vanishes in the limit. Hence 



T = pfff(Lt; + My + NQ dxdydz ............... (28). 



If we substitute the values of L, M, N from (11), this becomes 



(29), 



where each of the volume integrations extends over all the vortices. 



136. Under the same circumstances we have another useful 

 expression for T-, viz. 



T = 2p!ff{u (yt- zy) +v(z%- arf) +w(^- y)}dxdydz. . . (30). 



To verify this, we take the right-hand member, and transform it 

 by the process already so often employed, omitting the surface- 

 integrals for the same reason as in the preceding article. The first 

 of the three terms gives 



* They are in fact of the order 2 , as may be seen (for example) by calculating 



the value of L for a single closed vortex, and expressing it, by the method of Art. 

 133, as a surface-integral taken over a surface bounded by the vortex. Conse- 



,quently the velocities w, v, w are really of the order - . 



