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

70 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 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 and  is 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 This equation is equivalent to  or,