Page:PoyntingTransfer.djvu/5

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

(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 direction cosines of the normal to the surface outwards.