Page:Philosophical magazine 21 series 4.djvu/313

Rh surface be u, v, w. Then the work expended on that clement of surface will be

Let us begin with the first term, $$PudS$$. $$P$$ may be written

and

$u=n\beta-m\gamma.\,$

Remembering that the surface of the vortex is a closed one, so that

and

$\Sigma mydS=\Sigma nzdS=V,\,$

we find

and the whole work done on the vortex in unit of time will be

Prop. VIII.— To find the relations between the alterations 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

and

Comparing this value with that given in equation (50), we find

This equation being true for all values of $$\alpha,\beta$$, and $$\gamma$$, first let $$\beta$$ and $$\gamma$$ vanish, and divide by $$\alpha$$. We find