Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/77

This equation expresses the well-known theorem that if the geometrical sum of three vectors is zero, the magnitude of each vector is proportional to the sine of the angle between the other two. It also indicates the numerical coefficients by which one of three complanar vectors may be expressed in parts of the other two.

138. Def.—If two dyadics $\Phi$ and $\Psi$ are such that

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

they are said to be homologous.

If any number of dyadics are homologous to one another, and any other dyadics are formed from them by the operations of taking multiples, sums, differences, powers, reciprocals, or products, such dyadics will be homologous to each other and to the original dyadics. This requires demonstration only in regard to reciprocals. Now if

${\begin{aligned}\Phi .\Psi &=\Psi .\Phi ,\\\Psi .\Phi ^{-1}=\Phi ^{-1}.\Phi .\Psi .\Phi ^{-1}&=\Phi ^{-1}.\Psi .\Phi .\Phi ^{-1}=\Phi ^{-1}.\Psi .\end{aligned}}$

That is, $\Phi ^{-1}$ is homologous to $\Psi ,$ if $\Phi$ is.

134. If we call $\Psi .\Phi ^{-1}$ or $\Phi ^{-1}.\Psi$ the quotient of $\Psi$ and $\Phi ,$ we may say that the rules of addition, subtraction, multiplication and division of homologous dyadics are identical with those of arithmetic or ordinary algebra, except that limitations analogous to those respecting zero in algebra must be observed with respect to all incomplete dyadics.

It foUows that the algebraic and higher analysis of homologous dyadics is substantially identical with that of scalars.

135. It is always possible to express a dyadic in three terms, so that both the antecedents and the consequents shall be perpendicular among themselves.

To show this for any dyadic $\Phi ,$ let us set

$\rho '=\Phi .\rho ,$

$\rho$ being a unit-vector, and consider the different values of $\rho$ for all possible directions of $\rho .$ Let the direction of the unit vector $i$ be so determined that when $\rho$ coincides with $i,$ the value of $\rho '$ shall be at least as great as for any other direction of $\rho .$ And let the direction of the unit vector $j$ be so determined that when $\rho$ coincides with $j,$ the value of $\rho '$ shall be at least as great as for any other direction of $\rho$ which is perpendicular to $i.$ Let $k$ have its usual position with respect to $i$ and $j.$ It is evidently possible to express $\Phi$ in the form

$\alpha i+\beta j+\gamma k.$

We have therefore

$\rho '=\{\alpha i+\beta j+\gamma k\}.\rho ,$

and

$d\rho '=\{\alpha i+\beta j+\gamma k\}.d\rho .$