NOTES ON HYDRODYNAMICS, CHIEFLY ON VORTEX-MOTION. 15 



within a fixed closed surface. From the first form of the right-hand side of (36) 

 we obtain, since 



dx dy dz 



the result 



I I lj^dxdydz= I l(Zq n -wo> n )dS . . . (39), 



where q n is the normal velocity inwards along the normal at the element rfS of the 

 enclosing surface, and w n is the angular velocity about that normal. If the surface is 

 taken large enough to include all the vortices, £ and &>„ are zero at every element of the 

 surface, and we have 



f f f d ^dxdydz=0 .... (40). 



A similar result is obtained, of course, for each of the other components. The 

 volume-integral of the time-rate of change of each component of angular velocity 

 of spin has thus the value indicated by the right-hand side of (39), and vanishes for 

 any surface enclosing all the vortices. 



If we take the second form of the right-hand side of (36) we get 



fffKdzdydz= (Uq n dS - f f nmq cos ^dS - f j f (v d J + vp- + vB^dxdydz . (41), 



where n is the 2-direction cosine of the inward-drawn normal to ds, and <p is the 

 angle between the directions of q and (p at t/S. By (40) this gives the equation 



f ((ww,, - nwg cos </>)r/S = f f l(JL + v ^L+w d ^-\dxdydz . . (42). 



