ON PHYSICAL LINES OF FORCE. 473 



then fa Is the potential at any point due to the magnetic system m 1} and fa 

 that due to the distribution of magnetism represented by m,. The actual 

 energy of all the vortices is 



E^SCpp + F + tfdV ............................. (38), 



the integration being performed over all space. 



This may be shewn by integration by parts (see Green's ' Essay on Elec- 

 tricity/ p. 10) to be equal to 



E= -^(^(fam. + fam^fam^ + fam^dV ................. (39). 



Or since it has been proved (Green's 'Essay,' p. 10) that 



^ V, 



V'. ................. (40). 



Now let the magnetic system m l remain at rest, and let m, be moved 

 parallel to itself in the direction of x through a space So; ; then, since fa 

 depends on m^ only, it will remain as before, so that fan^ will be constant ; 

 and since fa depends on wj, only, the distribution of fa about m, will remain 

 the same, so that fam t will be the same as before the change. The only part 

 of E that will be altered is that depending on 2fam,, because fa becomes 



fa + -y- 1 8x on account of the displacement. The variation of actual energy due 

 to the displacement is therefore 



ax 



But by equation (12) the work done by the mechanical forces on m 3 during 

 the motion is 



(42); 



and since our hypothesis is a purely mechanical one, we must have by the 

 conservation of force, 



8E + 8W=0 ............................... (43); 



that is, the loss of energy of the vortices must be made up by work done in 

 moving magnets, so that 



2 m t dv Bx + 2 ' <m 3 d V 8x = 0, 

 \ ctx / \ct/x / 



c =sl ...................................... < 44 >- 



VOL. I. 60 



