288 Prof. Maxwell on the Theory of Molecular Vortices 
have by the conservation of force, 
sE+6W=0; . 4 2 3. 6 et 
that is, the loss of energy of the vortices must be made up by 
work done in moving magnets, so that 
— 403 (2! mdV Jb +3 (“P myaV )8x=0, 
or 
Cg cm ko yale 
~ Sar? 
so that the energy of the vortices in unit of volume is 
i 
gp Mla + B+") 5 » Mr te 
and that of a vortex whose volume is V is 
a Wet B+ yV. 0 
In order to produce or destroy this energy, work must be ex- 
pended on, or received from, the vortex, either by the tangential 
action of the layer of particles in contact with it, or by change 
of form in the vortex. We shall first investigate the tangential 
action between the vortices and the layer of particles in contact 
with them. 
Prop. VII.—To find the energy spent upon a vortex in unit of 
time by the layer of particles which surrounds it. 
Let P, Q, R be the forces acting on unity of the particles in 
the three coordinate directions, these quantities being functions 
of z, y, and z. Since each particle touches two vortices at the 
extremities of a diameter, the reaction of the particle on the vor- 
tices will be equally divided, and will be 
‘Ie ] ] 
2 ts 2 Q, 2 ; 
on each vortex for unity of the particles ; but since the superficial 
density of the particles is Bh (see equation (34)), the forces on 
Qe 
unit of surface of a vortex wili be 
eee ] ] 
nt gens mae — an 
Now let dS be an element of the surface of a vortex. Let the 
direction-cosines of the normal be /, m, x. Let the coordinates 
of the element be z, y, z. Let the component velocities of the 
