Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/75

127. Def.—We shall write $\Phi ^{-1}$ for the reciprocal of any (complete) dyadic $\Phi ,$ also $\Phi ^{2}$ for $\Phi .\Phi$ etc., and $\Phi ^{-2}$ for $\Phi ^{-1}.\Phi ^{-1}$ etc. It is evident that $\Phi ^{-n}$ is the reciprocal of $\Phi ^{n}$,

128. In the reduction of equations, if we have

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

we may cancel the $\Phi$ (which is equivalent to multiplying by $\Phi ^{-1}$) if $\Phi$ is a complete dyadic, but not otherwise The case is the same with such equations as

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

To cancel an incomplete dyadic in such cases would be analogous to cancelling a zero factor in algebra.

129. Def.—If in any dyadic we transpose the factors in each term, the dyadic thus formed is said to be conjugate to the first. Thus

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

are conjugate to each other. A dyadic of which the value is not altered by such transposition is said to be self-conjugate. The conjugate of any dyadic $\Phi$ may be written $\Phi _{\text{C}}$ It is evident that

$\rho .\Phi =\Phi _{\text{C}}.\rho$⁠and⁠$\Phi .\rho =\rho .\Phi _{\text{C}}.$

$\Phi _{\text{C}}.\rho$ and $\Phi .\rho$ are conjugate functions of $\rho .$ (See No. 106.) Since $\{\Phi _{\text{C}}\}^{2}=\{\Phi ^{2}\}_{\text{C}},$ we may write $\Phi _{\text{C}}^{2},$ etc., without ambiguity.

130. The reciprocal of the product of any number of dyadics is equal to the product of their reciprocals taken in inverse order. Thus

$\{\Phi .\Psi .\Omega \}_{\text{C}}=\Omega _{\text{C}}.\Psi _{\text{C}}.\Phi _{\text{C}}\}^{-1}=\Omega ^{-1}.\Psi ^{-1}.\Phi ^{-1}.$

The conjugate of the product of any number of dyadics is equal to the product of their conjugates taken in inverse order. Thus

$\{\Phi .\Psi .\Omega \}_{\text{C}}=\Omega _{\text{C}}.\Psi _{\text{C}}.\Phi _{\text{C}}.$

Hence, since

$\Phi _{\text{C}}.\{\Phi ^{-1}\}_{\text{C}}=\{\Phi ^{-1}.\Phi \}_{\text{C}}=I,$ $\{\Phi ^{-1}\}_{\text{C}}=\{\Phi _{\text{C}}\}^{-1},$

and we may write $\Phi _{\text{C}}^{-1}$ without ambiguity.

131. It is sometimes convenient to be able to express by a dyadic taken in direct multiplication the same operation which would be effected by a given vector ($\alpha$) in skew multiplication. The dyadic $I\times \alpha$ will answer this purpose. For, by No. 117,

$\{I\times \alpha \}.\rho =\alpha \times \rho ,$⁠	$\rho .\{I\times \alpha \}=\rho \times \alpha ,$
$\{I\times \alpha \}.\Phi =\alpha \times \Phi ,$	$\Phi .\{I\times \alpha \}=\Phi \times \alpha .$

The same is true of the dyadic $\alpha \times I,$ which is indeed identical with $I\times \alpha ,$ as appears from the equation $I.\{\alpha \times I\}=\Phi .\{I\times \alpha \}.I.$