Page:Philosophical magazine 21 series 4.djvu/308

${\begin{array}{lll}n\beta -m\gamma &&\mathrm {parallel\ to} \ x,\\l\gamma -n\alpha &&\mathrm {parallel\ to} \ y,\\m\alpha -l\beta &&\mathrm {parallel\ to} \ z.\end{array}}$

If this portion of the surface be in contact with another vortex whose velocities are $\alpha ',\beta ',\gamma '$, then a layer of very small particles placed between them will have a velocity which will be the mean of the superficial velocities of the vortices which they separate, so that if $u$ is the velocity of the particles in the direction of $x$,

$u={\frac {1}{2}}m(\gamma '-\gamma )-{\frac {1}{2}}n(\beta '-\beta ),$

(27)

since the normal to the second vortex is in the opposite direction to that of the first.

Prop. V.— To determine the whole amount of particles transferred across unit of area in the direction of $x$ in unit of time.

Let $x_{1},y_{1},z_{1}$ be the coordinates of the centre of the first vortex, $x_{2},y_{2},z_{2}$ those of the second, and so on. Let $V_{1},V_{2}$, &c. be the volumes of the first, second, &c. vortices, and ${\overline {V}}$ the sum of their volumes. Let $dS$ be an element of the surface separating the first and second vortices, and x, y, z its coordinates. Let $\rho$ be the quantity of particles on every unit of surface. Then if $p$ be the whole quantity of particles transferred across unit of area in unit of time in the direction of $x$, the whole momentum parallel to $x$ of the particles within the space whose volume is ${\overline {V}}$ will be ${\overline {V}}p$, and we shall have

${\overline {V}}p=\Sigma u\rho dS,$

(28)

the summation being extended to every surface separating any two vortices within the volume ${\overline {V}}$.

Let us consider the surface separating the first and second vortices. Let an element of this surface be $dS$, and let its direction-cosines be $l_{1},m_{1},n_{1}$ with respect to the first vortex, and $l_{2},m_{2},n_{2}$ with respect to the second; then we know that

$l_{1}+l_{2}=0,\ m_{1}+m_{2}=0,\ n_{1}+n_{2}=0.$

(29)

The values of $\alpha ,\beta ,\gamma$ vary with the position of the centre of the vortex; so that we may write

$\alpha _{2}=\alpha _{1}+{\frac {d\alpha }{dx}}\left(x_{2}-x_{1}\right)+{\frac {d\alpha }{dy}}\left(y_{2}-y_{1}\right)+{\frac {d\alpha }{dz}}\left(z_{2}-z_{1}\right),$

(30)

with similar equations for $\beta$ and $\gamma$.

The value of $u$ may be written:—