Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/73

120. In like manner, the sum of two dyads may be reduced to a single dyad, if either the antecedents or the consequents are parallel, and only in such cases.

A sum of three dyads cannot be reduced to a single dyad, unless either their antecedents or consequents are parallel, or both antecedents and consequents are (separately) complanar. In the first case the reduction can always be made, in the second, occasionally.

121. Def.—A dyadic which cannot be reduced to the sum of less than three dyads will be called complete.

A dyadic which can be reduced to the sum of two dyads will be called planar. When the plane of the antecedents coincides with that of the consequents, the dyadic will be called uniplanar. These planes are invariable for a given dyadic, although the dyadic may be so expressed that either the two antecedents or the two consequents may have any desired values (which are not parallel) within their planes.

A dyadic which can be reduced to a single dyad will be called linear. When the antecedent and consequent are parallel, it will be called unilinear.

A dyadic is said to have the value zero when all its terms vanish.

122. If we set

$\sigma =\Phi .\rho ,$⁠$\tau =\rho .\Phi ,$

and give $\rho$ all possible values, $\sigma$ and $\tau$ will receive all possible values, if $\Phi$ is complete. The values of $\sigma$ and $\tau$ will be confined each to a plane if $\Phi$ is planar, which planes will coincide if $\Phi$ is uniplanar. The values of $\sigma$ and $\tau$ will be confijied each to a line if $\Phi$ is linear, which lines will coincide if $\Phi$ is unilinear.

123. The products of complete dyadics are complete, of complete and planar dyadics are planar, of complete and linear dyadics are linear.

The products of planar dyadics are planar, except that when the plane of the consequents of the first dyadic is perpendicular to the plane of the antecedents of the second dyadic, the product reduces to a linear dyadic.

The products of linear dyadics are linear, except that when the consequent of the first is perpendicular to the antecedent of the second, the product reduces to zero.

The products of planar and linear dyadics are linear, except when, the planar preceding, the plane of its consequents is perpendicular to the antecedent of the linear, or, the linear preceding, its consequent is perpendicular to the plane of the antecedents of the planar. In these cases the product is zero.

All these cases are readily proved, if we set

$\sigma =\Phi .\Psi .\rho ,$