Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/388

346 If we were at liberty to assume that a solid body may be treated as a system of linear conductors, then, from the reciprocal property (Art. 281) of any two conductors of a linear system, we might shew that the electromotive force along $$z$$ required to produce a unit current parallel to $$y$$ must be equal to the electromotive force along $$y$$ required to produce a unit current parallel to $$z.$$ This would shew that $$P_1 = Q_l,$$ and similarly we should find $$ P_2 = Q_2,$$ and $$P_3 = Q_3.$$ When these conditions are satisfied the system of coefficients is said to be Symmetrical. When they are not satisfied it is called a Skew system.

We have great reason to believe that in every actual case the system is symmetrical, but we shall examine some of the consequences of admitting the possibility of a skew system.

298.] The quantities $$u,\, v,\, w$$ may be expressed as linear functions of $$X,\, Y,\, Z$$ by a system of equations, which we may call Equations of Conductivity,{{numb form| $$ \left. \begin{matrix} u & = &r_1X+p_3Y+q_2Z, \\ v & = & q_3X+r_2Y+p_1Z, \\ w & =& p_2X+q_1Y+r_3Z; \end{matrix} \right \} $$ | (3)}}we may call the coefficients $$r$$ the coefficients of Longitudinal conductivity, and $$ p$$ and $$q $$ those of Transverse conductivity.

The coefficients of resistance are inverse to those of conductivity. This relation may be defined as follows:

Let $$[PQR]$$ be the determinant of the coefficients of resistance, and $$[pqr]$$ that of the coefficients of conductivity, then

The other equations may be formed by altering the symbols $$ P,\, Q,\, R,\, p,\, q,\, r,$$ and the suffixes $$1,\, 2,\, 3$$ in cyclical order.

Rate of Generation of Heat.

299.] To find the work done by the current in unit of time in overcoming resistance, and so generating heat, we multiply the components of the current by the corresponding components of the electromotive force. We thus obtain the following expressions for $$W,$$ the quantity of work expended in unit of time: