Page:Philosophical magazine 21 series 4.djvu/313

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

${\frac {dE}{dt}}=-{\frac {1}{4\pi }}(Pu+Qv+Rw)dS.$

(47)

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

$P_{0}+{\frac {dP}{dx}}x+{\frac {dP}{dy}}y+{\frac {dP}{dz}}z,$

(48)

and

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

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

$\Sigma nxdS=\Sigma mxdS=\Sigma nydS=\Sigma mzdS=0,\,$

and

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

we find

$\Sigma PudS=\left({\frac {dP}{dz}}\beta -{\frac {dP}{dy}}\gamma \right)V,$

(49)

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

${\begin{array}{l}{\frac {dE}{dt}}=-{\frac {1}{4\pi }}\Sigma (Pu+Qv+Rw)dS\\\\={\frac {1}{4\pi }}\left\{\alpha \left({\frac {dQ}{dz}}-{\frac {dR}{dy}}\right)+\beta \left({\frac {dR}{dx}}-{\frac {dP}{dz}}\right)+\gamma \left({\frac {dP}{dy}}-{\frac {dQ}{dx}}\right)\right\}V.\end{array}}$

(50)

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

$E={\frac {1}{8\pi }}\mu \left(\alpha ^{2}+\beta ^{2}+\gamma ^{2}\right)V,$

(51)

and

${\frac {dE}{dt}}={\frac {1}{4\pi }}\mu V\left(\alpha {\frac {d\alpha }{dt}}+\beta {\frac {d\beta }{dt}}+\gamma {\frac {d\gamma }{dt}}\right).$

(52)

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

$\alpha \left({\frac {dQ}{dz}}-{\frac {dR}{dy}}-\mu {\frac {d\alpha }{dt}}\right)+\beta \left({\frac {dR}{dx}}-{\frac {dP}{dz}}-\mu {\frac {d\beta }{dt}}\right)+\gamma \left({\frac {dP}{dy}}-{\frac {dQ}{dx}}-\mu {\frac {d\gamma }{dt}}\right)=0.$

(53)

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