Page:PoyntingTransfer.djvu/5

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\left.\begin{array}{c}
P=c\dot{y}+b\dot{z}-\frac{dF}{dt}-\frac{d\psi}{dx}=c\dot{y}+b\dot{z}+P'\\
\\Q=a\dot{z}+c\dot{x}-\frac{dG}{dt}-\frac{d\psi}{dy}=a\dot{z}+c\dot{x}+Q'\\
\\R=b\dot{x}+a\dot{y}-\frac{dH}{dt}-\frac{d\psi}{dz}=b\dot{x}+a\dot{y}+R'\end{array}\right\} (5)

where P', Q', R' are put for the parts of P, Q, R which do not contain the velocities.

Then

\begin{array}{ll}
Pu+Qv+Rw & =(c\dot{y}-b\dot{z})u+(a\dot{z}-c\dot{x})v+(b\dot{x}-a\dot{y})w+P'u+Q'v+R'w\\
\\ & =-\left\{ (vc-wb)\dot{x}+(wa-uc)\dot{y}+(ub-va)\dot{z}\right\} +P'u+Q'v+R'w\\
\\ & =-(X\dot{x}+Y\dot{y}+Z\dot{z})+P'u+Q'v+R'w,\end{array}

where X, Y, Z are the components of the electromagnetic force per unit of volume (Maxwell, vol. ii, p. 227).

Now substituting in (4) and putting for u, v, w their values in terms of the magnetic force (Maxwell, vol. ii, p. 233) and transposing we obtain

\begin{array}{l}
\frac{K}{4\pi}\iiint\left(P\frac{dP}{dt}+Q\frac{dQ}{dt}+R\frac{dR}{dt}\right)dx\ dy\ dz+\iiint\left\{ (X\dot{x}+Y\dot{y}+Z\dot{z})+(Pp+Qq+Rr)\right\} dx\ dy\ dz\\
\\\qquad=\iiint(P'u+Q'v+R'w)dx\ dy\ dz\\
\\\qquad=\frac{1}{4\pi}\iiint\left\{ P'\left(\frac{d\gamma}{dy}-\frac{d\beta}{dz}\right)+Q'\left(\frac{d\alpha}{dz}-\frac{d\gamma}{dx}\right)+R'\left(\frac{d\beta}{dx}-\frac{d\alpha}{dy}\right)\right\} dx\ dy\ dz\\
\\\qquad=\frac{1}{4\pi}\iiint\left(R'\frac{d\beta}{dx}-Q'\frac{d\gamma}{dx}\right)dx\ dy\ dz\\
\\\qquad+\frac{1}{4\pi}\iiint\left(P'\frac{d\gamma}{dy}-R'\frac{d\alpha}{dy}\right)dx\ dy\ dz\\
\\\qquad+\frac{1}{4\pi}\iiint\left(Q'\frac{d\alpha}{dz}-P'\frac{d\beta}{dz}\right)dx\ dy\ dz\end{array}

[Integrating each term by parts)]

\begin{array}{c}
=\frac{1}{4\pi}\iint(R'\beta-Q'\gamma)dy\ dz+\frac{1}{4\pi}\iint(P'\gamma-R'\alpha)dz\ dx+\frac{1}{4\pi}\iint(Q'\alpha-P'\beta)dx\ dy\\
\\-\frac{1}{4\pi}\iiint\left\{ \beta\frac{dR'}{dx}-\gamma\frac{dQ'}{dx}+\gamma\frac{dP'}{dy}-\alpha\frac{dR'}{dy}+\alpha\frac{dQ'}{dz}-\beta\frac{dP'}{dz}\right\} dx\ dy\ dz\end{array}

(The double integral being taken over the surface)

\begin{array}{c}
=\frac{1}{4\pi}\iint\left\{ l(R'\beta-Q'\gamma)+m(P'\gamma-R'\alpha)+n(Q'\alpha-P'\beta)\right\} dS\\
\\-\frac{1}{4\pi}\iiint\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\} dx\ dy\ dz\end{array} (6)

where l, m, n are the directioncosines of the normal to the surface outwards.