Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/86

where $\alpha ',\beta ',\gamma '$ are the reciprocals of $\alpha ,\beta ,\gamma ,$ and $a,b,c,p,$ and $q$ are scalars, of which $p$ is positive, will be most evident if we resolve it into the factors

$a\alpha \alpha '+\beta \beta '+\gamma \gamma ',$ $\alpha \alpha '+p\beta \beta '+p\gamma \gamma ',$ $\alpha \alpha '+\cos q\{\beta \beta '+\gamma \gamma \}+\sin q\{\gamma \beta '-\beta \gamma '\},$

of which the order is immaterial, and if we suppose the vector on which we operate to be resolved into two factors, one parallel to $\alpha ,$ and the other in the $\beta {\text{-}}\gamma$ plane. The effect of the first factor is to multiply by $a$ the component parallel to $\alpha ,$ without affecting the other. The effect of the second is to multiply by $p$ the component in the $\beta {\text{-}}\gamma$ plane without affecting the other. The effect of the third is to give the component in the $\beta {\text{-}}\gamma$ plane the kind of elliptic rotation described in No. 147.

The effect of the same dyadic as a postfactor is of the same nature.

The value of the dyadic is not affected by the substitution for a of another vector having the same direction, nor by the substitution for $\beta$ and $\gamma$ of two other conjugate semi-diameters of the same or a similar and similarly situated ellipse, and which follow one another in the same angular direction.

Def. — Such dyadics we shall call cyclotonic.

154. Cyclotonics which are reducible to the same form except with respect to the values of $a,p,$ and $q$ are homologous. They are multiplied by multiplying the values of $a,$ and also those of $p,$ and adding those of $q.$ Thus, the product of

$a_{1}\alpha \alpha '+p_{1}\,\cos q_{1}\{\beta \beta '+\gamma \gamma '\}+p_{1}\,\sin q_{1}\{\gamma \beta '-\beta \gamma '\}$

and

$a_{2}\alpha \alpha '+p_{2}\,\cos q_{2}\{\beta \beta '+\gamma \gamma '\}+p_{2}\,\sin q_{2}\{\gamma \beta '-\beta \gamma '\}$

is

${\begin{aligned}a_{1}a_{2}\alpha \alpha '+p_{1}p_{2}\cos(q_{1}&+q_{2})\{\beta \beta '+\gamma \gamma '\}\\&+p_{1}p_{2}\sin(q_{1}+q_{2})\{\gamma \beta '-\beta \gamma '\}\end{aligned}}$

A dyadic of this form, in which the value of $q$ is not zero, or the product of $\pi$ and a positive or negative integer, is homologous only with such dyadics as are obtained by varying the values of $a,p,$ and $q.$

156. In general, any dyadic may be reduced to the form either of a tonic or of a cyclotonic. (The exceptions are such as are made by the limiting cases.) We may show this, and also indicate how the reduction may be made, as follows. Let $\Phi$ be any dyadic. We have first to show that there is at least one direction of $\rho$ for which

$\Phi .\rho =a\rho .$

This equation is equivalent to

$\Phi .\rho -a\rho =0,$

or,

$\{\Phi -a{\text{I}}\}.\rho =0.$