Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/389

$W=Xu+Yv+Zw;$

(8)

$=R_{1}u^{2}+R_{2}v^{2}+R_{3}w^{2}+(P_{1}+Q_{1})vw+(P_{2}+Q_{2})wu+(P_{3}+Q_{3})uv;$

(9)

$=r_{1}X^{2}+r_{2}Y^{2}+r_{3}Z^{2}+(p_{1}+q_{1})YZ+(p_{2}+q_{2})ZX+(p_{3}+q_{3})XY.$

(10)

By a proper choice of axes, either of the two latter equations may be deprived of the terms involving the products of $u,v,w$ or of $X,Y,Z.$ The system of axes, however, which reduces $W$ to the form

$R_{1}u^{2}+R_{2}v^{2}+R_{3}w^{2}$

is not in general the same as that which reduces it to the form

$r_{1}X^{2}+r_{2}Y^{2}+r_{3}Z^{2}.$

It is only when the coefficients $P_{1},P_{2},P_{3}$ are equal respectively to $Q_{1}Q_{2},Q_{3}$ that the two systems of axes coincide.

If with Thomson^[1] we write

and

$\left.{\begin{matrix}P&=&S+T,&\quad &Q&=&S-T;\\p&=&s+t,&\quad &q&=&s-t;\end{matrix}}\right\}$

(11)

then we have

$\left.{\begin{array}{rcl}[PQR]&=&R_{1}R_{2}R_{3}+2S_{1}S_{2}S_{3}-S_{1}^{2}R_{1}-S_{2}^{2}R_{2}-S_{3}^{2}R_{3}\\&&+2(S_{1}T_{2}T_{3}+S_{2}T_{3}T_{1}+S_{3}T_{1}T_{2})+R_{1}T_{1}^{2}+R_{2}T_{2}^{2}+R_{3}T_{3}^{2};\end{array}}\right\}$

(12)

and

${\begin{array}{rcl}[PQR]r_{1}&=&R_{2}R_{3}-S_{1}^{2}+T_{1}^{2},\\\left[PQR\right]s_{1}&=&T_{2}T_{3}+S_{2}S_{3}-R_{1}S_{1},\\\left[PQR\right]t_{1}&=&-R_{1}T_{1}+S_{2}T_{3}+S_{3}T_{2}.\end{array}}$

$\left.{\begin{matrix}\\\\\\\end{matrix}}\right\}$(13)

If therefore we cause $S_{l},S_{2},S_{3}$ to disappear, $s_{1}$ will not also disappear unless the coefficients $T$ are zero.

Condition of Stability.

300.] Since the equilibrium of electricity is stable, the work spent in maintaining the current must always be positive. The conditions that $W$ must be positive are that the three coefficients $R_{1},R_{2},R_{3},$ and the three expressions

$\left.{\begin{matrix}4R_{2}R_{3}&-&(P_{1}+Q_{1})^{2},\\4R_{3}R_{1}&-&(P_{2}+Q_{2})^{2},\\4R_{1}R_{2}&-&(P_{3}+Q_{3})^{2},\end{matrix}}\right\}$

(14)

must all be positive.

There are similar conditions for the coefficients of conductivity.

1. Trans. R. S. Edin., 1853-4, p. 165.