Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/391

of the form $A$ will be the reciprocal of the corresponding coefficient of the form $r.$ The expression for $\rho$ will be

${\frac {x^{2}}{r_{1}}}+{\frac {y^{2}}{r_{2}}}+{\frac {z^{2}}{r_{3}}}={\frac {\rho ^{2}}{r_{1}r_{2}r_{3}}}.$

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303.] The theory of the complete system of equations of resistance and of conductivity is that of linear functions of three variables, and it is exemplified in the theory of Strains^[1], and in other parts of physics. The most appropriate method of treating it is that by which Hamilton and Tait treat a linear and vector function of a vector. We shall not, however, expressly introduce Quaternion notation.

The coefficients $T_{1},T_{2},T_{3}$ may be regarded as the rectangular components of a vector $T,$ the absolute magnitude and direction of which are fixed in the body, and independent of the direction of the axes of reference. The same is true of $t_{1},t_{2},t_{3},$ which are the components of another vector $t.$

The vectors $T$ and $t$ do not in general coincide in direction.

Let us now take the axis of $z$ so as to coincide with the vector $T,$ and transform the equations of resistance accordingly. They will then have the form

$\left.{\begin{array}{rcl}X&=&R_{1}u+S_{3}v+S_{2}w-Tv,\\Y&=&S_{3}u+R_{2}v+S_{1}w+Tu,\\Z&=&S_{2}u+S_{1}v+R_{3}w.\end{array}}\right\}$

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It appears from these equations that we may consider the electromotive force as the resultant of two forces, one of them depending only on the coefficients $R$ and $S,$ and the other depending on $T$ alone. The part depending on $R$ and $S$ is related to the current in the same way that the perpendicular on the tangent plane of an ellipsoid is related to the radius vector. The other part, depending on $T,$ is equal to the product of $T$ into the resolved part of the current perpendicular to the axis of $T$, and its direction is perpendicular to $T$ and to the current, being always in the direction in which the resolved part of the current would lie if turned 90° in the positive direction round $T.$

Considering the current and $T$ as vectors, the part of the electromotive force due to $T$ is the vector part of the product, $T\times \mathrm {current.}$

The coefficient $T$ may be called the Rotatory coefficient. We

1. * See Thomson and Tait's Natural Philosophy § 154.