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

300.]

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

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 we write {{numb form| $$ \left. \begin{matrix} P& = & S+T, & \quad &Q &= &S-T; \\p & = & s+t, & \quad &q & = &s-t; \end{matrix} \right \} $$ |(11) |and }} then we have {{numb form| $$ \left. \begin{array}{rcl} [PQR] & = & R_1R_2R_3 + 2S_1S_2S_3 - S_1^2R_1 - S_2^2R_2 - S_3^2R_3 \\ & & + 2(S_1T_2T_3+S_2T_3T_1+S_3T_1T_2) +R_1T_1^2 +R_2T_2^2 +R_3T_3^2; \end{array} \right \} $$ |(12) }} {{numb form| $$ \begin{array}{rcl} [PQR]r_1 & = & R_2 R_3 - S_1^2+T_1^2, \\ \left [ PQR \right ] s_1 & = & T_2 T_3 + S_2 S_3 - R_1 S_1, \\ \left [ PQR \right ] t_1 & = & -R_1 T_1+S_2 T_3+S_3 T_2. \end{array} $$ | $$ \left. \begin{matrix} \\ \\ \\ \end{matrix} \right \} $$(13) | and }}

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 {{numb form| $$ \left. \begin{matrix} 4R_2R_3 & - & (P_1+Q_1)^2, \\ 4R_3R_1 & - & (P_2+Q_2)^2, \\ 4R_1R_2 &-&(P_3+Q_3)^2, \end{matrix} \right \} $$ | (14)}} must all be positive.

There are similar conditions for the coefficients of conductivity.