applied to Electric Currents. 287 
in unit of volume is 
E=Cy(a?+ 6?+ 97), 
where C is a constant to be determined. 
Let us take the case in which 
ae pee ae 
“Tle? B= dy’ Y= ‘ge’ e ° . (35) 
Let 
b=, + do, sign aM hoy ie 
and let 
— 
“all d*h, d*, oS) =m, and —— ‘iad + (oo d*ho =i) = Mp} ; 37) 
dem \ dx? * “dy? du? * “dy? * de? 
then ¢, is the potential at any point due to A magnetic system 
m,, and ¢, that due to the distribution of magnetism represented 
by m,. The actual energy of all the vortices is 
B=SCy(e2+62+7%)dV, . . . . (88) 
the integration being performed over all space. 
This may be shown by integration by parts (see Green’s 
‘Essay on Electricity,’ p. 10) to be equal to 
= —4arCX (p,m, + Poma+ Pymat+ghym)dV. . (89) 
Or since it has been proved (Green’s ‘ Essay,’ p. 10) that 
d,m dV =Xidh.m,dV, 
E= —4rC(gym, + dgmgt+2hym,)dV. . . . (40) 
Now let the magnetic system m, remain at rest, and let m, be 
moved parallel to itself in the direction of # through a space dz ; 
then, since ¢, depends on m, only, it will remain as before, so 
that $m, will be constant; and since ¢, depends on m, only, 
the distribution of ¢, about m, will remain the same, so that 
go mz Will be the same as before the change. The only part of 
E that will be altered is that depending on 2¢,m,, because ¢, 
becomes ¢, + = dz on account of the displacement. The varia- 
tion of actual energy due to the displacement is therefore 
sE= —4rC> (2 ma) GV 02..." Nae) 
But by equation (12), the work done by the mechanical forces 
on m, during the motion is 
sw= = (Pima V)8e 5 ee ae 
and since our hypothesis is a purely mechanical one, we must 
