Page:On Faraday's Lines of Force.pdf/18

where

$4T^{2}=(Q_{1}-R_{1})^{2}+(Q_{2}-R_{2})^{2}+(Q_{3}-R_{3})^{2}$,

and l, m, n are direction-cosines of a certain fixed line in space.

The equations then become

$a=P_{1}\alpha +S_{3}\beta +S_{2}\gamma +(n\beta -m\gamma )T$,
$b=P_{2}\beta +S_{1}\gamma +S_{3}\alpha +(l\gamma -n\alpha )T$,
$c=P_{3}\gamma +S_{2}\alpha +S_{1}\beta +(m\alpha -l\beta )T$.

By the ordinary transformation of co-ordinates we may get rid of the coefficients marked S. The equations then become

$a=P_{1}'\alpha +(n'\beta -m'\gamma )T$,
$b=P_{2}'\beta +(l'\gamma -n'\alpha )T$,
$c=P_{3}'\gamma +(m'\alpha -l'\beta )T$,

where $l',m',n'$ are the direction-cosines of the fixed line with reference to the new axes. If we make

$\alpha ={\frac {dp}{dx}},\;\beta ={\frac {dp}{dy}},\;and\;\gamma ={\frac {dp}{dz}},$

the equation of continuity

${\frac {da}{dx}}+{\frac {db}{dy}}+{\frac {dc}{dz}}=0,$

becomes

$P_{1}'{\frac {d^{2}p}{dx^{2}}}+P_{2}'{\frac {d^{2}p}{dy^{2}}}+P_{3}'{\frac {d^{2}p}{dz^{2}}}=0,$

and if we make

$x={\sqrt {P_{1}'\xi }},\;\;y={\sqrt {P_{2}'\eta }},\;\;z={\sqrt {P_{3}'\zeta }},$

then

${\frac {d^{2}p}{d\xi ^{2}}}+{\frac {d^{2}p}{d\eta ^{2}}}+{\frac {d^{2}p}{d\zeta ^{2}}}=0,$

the ordinary equation of conduction.

It appears therefore that the distribution of pressures is not altered by the existence of the coefficient T. Professor Thomson has shewn how to conceive a substance in which this coefficient determines a property having reference to an axis, which unlike the axes of $P_{1},P_{2},P_{3}$ is dipolar.

For further information on the equations of conduction, see Professor Stokes On the Conduction of Heat in Crystals (Cambridge and Dublin Math. Journ.), and Professor Thomson On the Dynamical Theory of Heat, Part v. (Transactions of Royal Society of Edinburgh, VOL XXI. Part I.).