Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/412

of conductivity, Art. 298. If we put $D$ for the determinant of the coefficients of resistance, we find

${\begin{array}{ll}ur_{3}D&=R_{2}{\overline {X}}-Q_{3}{\overline {Y}}+{\overline {w}}q_{2}D,\\vr_{3}D&=R_{1}{\overline {Y}}-P_{3}{\overline {X}}+{\overline {w}}p_{1}D,\\Zr_{3}&=-p_{2}{\overline {X}}-q_{1}{\overline {Y}}+{\overline {w}}.\end{array}}$

Similar equations with the symbols accented give the values of $u',\,v'$ and $z'.$ Having found ${\overline {u}},\,{\overline {v}}$ and ${\overline {w}}$ in terms of ${\overline {X}},\,{\overline {Y}}$ and ${\overline {Z}},$ we may write down the equations of conductivity of the stratified conductor. If we make $h={\frac {c}{r_{3}}}$ and $h'={\frac {c'}{r_{3}'}},$ we find

${\begin{array}{l}{\overline {p}}_{1}={\frac {hp_{1}+h'p_{1}'}{h+h'}},\quad \quad {\overline {q}}_{1}={\frac {hq_{1}+h'q_{1}'}{h+h'}},\\{\overline {p}}_{2}={\frac {hp_{2}+h'p_{2}'}{h+h'}},\quad \quad {\overline {q}}_{2}={\frac {hq_{2}+h'q_{2}'}{h+h'}},\\{\overline {p}}_{3}={\frac {cp_{3}+c'p_{3}'}{c+c'}}-{\frac {hh'(q_{1}-q_{1}')(q_{2}-q_{2}')}{(h+h')(c+c')}},\\{\overline {q}}_{3}={\frac {cq_{3}+c'q_{3}'}{c+c'}}-{\frac {hh'(p_{1}-p_{1}')(p_{2}-p_{2}')}{(h+h')(c+c')}},\\{\overline {r}}_{1}={\frac {cr_{1}+c'r_{1}'}{c+c'}}-{\frac {hh'(p_{2}-p_{2}')(q_{2}-q_{2}')}{(h+h')(c+c')}},\\{\overline {r}}_{2}={\frac {cr_{2}+c'r_{2}'}{c+c'}}-{\frac {hh'(p_{1}-p_{1}')(q_{1}-q_{1}')}{(h+h')(c+c')}},\end{array}}$

${\begin{array}{c}{\overline {r}}_{3}={\frac {c+c'}{h+h'}}.\end{array}}$

320.] If neither of the two substances of which the strata are formed has the rotatory property of Art. 303, the value of any $P$ or $p$ will be equal to that of its corresponding $Q$ or $q.$ From this it follows that in the stratified conductor also

${\overline {p}}_{1}={\overline {q}}_{1},\quad \quad {\overline {p}}_{2}={\overline {q}}_{2},\quad \quad {\overline {p}}_{3}={\overline {q}}_{3},$

or there is no rotatory property developed by stratification, unless it exists in the materials.

321.] If we now suppose that there is no rotatory property, and also that the axes of $x,\,y$ and $z$ are the principal axes, then the $p$ and $q$ coefficients vanish, and

${\overline {r}}_{1}={\frac {cr_{1}+c'r_{1}'}{c+c'}},\quad \quad {\overline {r}}_{2}={\frac {cr_{2}+c'r_{2}'}{c+c'}},\quad \quad {\overline {r}}_{3}={\frac {c+c'}{{\frac {c}{r_{3}}}+{\frac {c'}{r_{3}'}}}}$