applied to Electric Currents. 289 
surface be u, v, w. Then the work expended on that element 
_ of surface will be 
Te =—X (Put Qt Rw)d8. . plank Sa 
Let us begin with the first term, PudS. P may be written 
ahr ge . BP 
Pot 7 # Ty as =a dz ay . . e ° ° e (48) 
and 
u=nB—my. 
Remembering that the surface of the vortex is a closed one, so 
that 
LaxdS ==Xmed8 = nydS = XmzdS=0, 
and 
LmydS = Znzd8S =V, 
we find 
dP dP 
SPudS = (—s- aa ME ieeareasarge so 
and the whole work done on the vortex in unit of time will be 
- =— = —=(Pu-+ Qu + Rw)dS 
3 = dR ees ui ak 2) 
=e = de dz)* Ve?) 
Prop. Be oy. find the relations Sais ee ee of 
motion of the vortices, and the forces P, Q, R which they exert 
on the layer of particles between them. 
Let V be the volume of a vortex, then by (46) its energy is 
i 
K= gn Met B+ 7*)V, UP aa eS FE 
and 
oe ied. oe 72). 
WE. age! Ns = tO tT ney 
Comparing this value with that given in equation (50), we find 
dQ dR _ pi) Pee ee dg 
“\de dy at) tO Nae ae ee 
dBi dQ dy\ _ P 
+7 ae cae: ==) She ee We eee (53) 
This equation being true for all values of a, 8, and y, first let 
8 and y vanish, and divide by «. We find 
Phil. Mag. 8. 4. Vol. 21. No. 140. April 1861. U 
