Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/415

324.] To construct a framework of linear conductors which shall have any given coefficients of conductivity forming a symmetrical system.

Fig. 24.

Let the space be divided into equal small cubes, of which let the figure represent one. Let the coordinates of the points $0,\,L,\,M,\,N,$ and their potentials be as follows:

${\begin{array}{lllll}\quad \quad &x&y&z\quad &Potential.\\O&0&0&0&0\\L&0&1&1&0+Y+Z,\\M&1&0&1&0+Z+X,\\N&1&1&0&0+X+Y.\end{array}}$

Let these four points be connected by six conductors,

$OL,\quad \quad OM,\quad \quad ON,\quad \quad MN,\quad \quad NL,\quad \quad LM,$

of which the conductivities are respectively

$A,\quad \quad B,\quad \quad C,\quad \quad P,\quad \quad Q,\quad \quad R.$

The electromotive forces along these conductors will be

$Y+Z,\quad \quad Z+X,\quad \quad X+Y,\quad \quad Y-Z,\quad \quad Z-X,\quad \quad X-Y,$

and the currents

$A(Y+Z),\quad B(Z+X),\quad C(X+Y),\quad P(Y-Z),\quad Q(Z-X),\quad R(X-Y).$

Of these currents, those which convey electricity in the positive direction of $x$ are those along $LM,\,LN,\,OM$ and $ON,$ and the quantity conveyed is

$u=(B+C+Q+R)X+(C-R)Y\quad \quad +(B-Q)Z.$

Similarly

${\begin{array}{lll}v=(C-R)X&+(C+A+R+P)Y&+(A-P)Y,\\w=(B-Q)X&+(A-P)Y&+(A+B+P+Q)Z;\end{array}}$

whence we find by comparison with the equations of conduction, Art. 298,

${\begin{array}{ll}4A=r_{2}+r_{3}-r_{1}+2p_{1},&4P=r_{2}+r_{3}-r_{1}-2p_{1}\\4B=r_{3}+r_{1}-r_{2}+2p_{2},&4Q=r_{3}+r_{1}-r_{2}-2p_{2}\\4C=r_{1}+r_{2}-r_{3}+2p_{3},&4R=r_{1}+r_{2}-r_{3}-2p_{3}\end{array}}$