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

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

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, 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 {{numb form| $$ \left. \begin{array}{rcl} X & = & R_1u+S_3v+S_2w -Tv, \\ Y & = & S_3u+R_2v+S_1w+Tu, \\ Z & = &S_2u+S_1v+R_3w. \end{array} \right \} $$ |(23)}}

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