Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/74

and consider the limits within which $\sigma$ varies, when we give $\rho$ all possible values.

The products $\Psi \times \rho$ and $\rho \times \Phi$ are evidently planar dyadics.

124. Def.—A dyadic $\Phi$ is said to be an idemfactor, when

$\Phi .\rho =\rho$⁠ for all values of $\rho ,$

or when

$\rho .\Phi =\rho$⁠ for all values of $\rho .$

If either of these conditions holds true, $\Phi$ must be reducible to the form

$ii+jj+kk.$

Therefore, both conditions will hold, if either does. All such dyadics are equal, by No. 108. They will be represented by the letter $I.$

The direct product of an idemfactor with another dyadic is equal to that dyadic. That is,

$I.\Phi =\Phi ,$⁠$\Phi .I=\Phi ,$

where $\Phi$ is any dyadic.

A dyadic of the form

$\alpha \alpha '+\beta \beta '+\gamma \gamma ',$

in which $\alpha ',\beta ',\gamma '$ are the reciprocals of $\alpha ,\beta ,\gamma ,$ is an idemfactor. (See No. 38.) A dyadic trinomial cannot be an idemfactor, unless its antecedents and consequents are reciprocals.

125. If one of the direct products of two dyadics is an idemfactor, the other is also. For, if $\Phi .\Psi =I,$

$\sigma .\Phi .\Psi =\sigma$

for all values of $\sigma ,$ and $\Phi$ is complete;

$\sigma .\Phi .\Psi .\Phi =\sigma .\Phi$

for all values of $\sigma ,$ therefore for all values of $\sigma .\Phi ,$ and therefore $\Psi .\Phi =I.$

Def. — In this case, either dyadic is called the reciprocal of the other.

It is evident that an incomplete dyadic cannot have any (finite) reciprocal.

Reciprocals of the same dyadic are equal. For if $\Phi$ and $\Psi$ are both reciprocals of $\Omega ,$

$\Phi =\Phi .\Omega .\Psi =\Psi .$

If two dyadics are reciprocals, the operators formed by using these dyadics as prefactors are inverse, also the operators formed by using them as postfactors.

126. The reciprocal of any complete dyadic

$\alpha \lambda +\beta \mu +\gamma \nu$

is

$\lambda '\alpha '+\mu '\beta '+\nu '\gamma ',$

where $\alpha ',\beta ',\gamma '$ are the reciprocals of $\alpha ,\beta ,\gamma ,$ and $\lambda ',\mu ',\nu '$ are the reciprocals of $\lambda ,\mu ,\nu .$ (See No. 38.)