470 ON PHYSICAL LINES OF FORCE. 



the direction of x, the whole momentum parallel to x of the particles within 

 the space whose volume is V will be Vp, and we shall have 



Vp^tupdS ................................. (28), 



the summation being extended to every surface separating any two vortices 

 within the volume V. 



Let us consider the surface separating the first and second vortices. Let an 

 element of this surface be dS, and let its direction-cosines be / m,, n, with 

 respect to the first vortex, and Z,, m^, n, with respect to the second; then we 

 know that 



Z, + Z, = 0, m, + OT 1 = 0, ni + n, = ..................... (29). 



The values of a, & y vary with the position of the centre of the vortex ; 

 so that we may write 



a , * (la , . da , 



with similar equations for /3 and y. 



The value of u may be written : 



u = i {m, (x - x t ) + m, (a; -a;,)} 



In effecting the summation of 2,updS, we must remember that round any 

 closed surface ZldS and all similar terms vanish; also that terms of the form 

 SlydS, where / and y are measured in different directions, also vanish ; but that 

 terms of the form tlxdS, where I and x refer to the same axis of co-ordinates, 

 do not vanish, but are equal to the volume enclosed by the surface. The 

 result is 



