Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/69

106. An expression of the form

$\alpha \,\lambda .\rho +\beta \,\mu .\rho +{\text{etc.}}$

evidently represents a linear function of $\rho ,$ and may be conveniently written in the form

$\{\alpha \lambda +\beta \mu +{\text{etc.}}\}.\rho .$

The expression $\rho .\alpha \lambda +\rho .\beta \mu +{\text{etc.,}}$

or$\rho .\{\alpha \lambda +\beta \mu +{\text{etc.}}\},$

also represents a linear function of $\rho ,$ which is, in general, different from the preceding, and will be called its conjugate.

107. Def.—An expression of the form $\alpha \lambda$ or $\beta \mu$ will be called a dyad. An expression consisting of any number of dyads united by the signs $+$ or $-$ will be called a dyadic binomial, trinomial, etc, as the case may be, or more briefly, a dyadic. The latter term will be used so as to include the case of a single dyad. When we desire to express a dyadic by a single letter, the Greek capitals will be used, except such as are like the Roman, and also $\Delta$ and $\mathrm {E} .$ The letter ${\text{I}}$ will also be used to represent a certain dyadic, to be mentioned hereafter.

Since any linear vector function may be expressed by means of a dyadic (as we shall see more particularly hereafter, see No. 110), the study of such functions, which is evidently of primary importance in the theory of vectors, may be reduced to that of dyadics.

108. Def.—Any two dyadics $\Phi$ and $\Psi$ are equal,

when	$\Phi .\rho =\Psi .\rho$	for all values of $\rho ,$
or, when	$\rho .\Phi =\rho .\Psi$	for all values of $\rho ,$
or, when	$\sigma .\Phi .\rho =\sigma .\Psi .\rho$	for all values of $\sigma$ and of $\rho .$

The third condition is easily shown to be equivalent both to the first and to the second. The three conditions are therefore equivalent.

It follows that $\Phi =\Psi ,$ if $\Phi .\rho =\Psi .\rho ,$ or $\rho .\Phi =\rho .\Psi ,$ for three non-complanar values of $\rho .$

109. Def.—We shall call the vector $\Phi .\rho$ the (direct) product of $\Phi$ and $\rho ,$ the vector $\rho .\Phi$ the (direct) product of $\rho$ and $\Phi ,$ and the scalar $\sigma .\Phi .\rho$ the (direct) product of $\sigma ,\Phi ,$ and $\rho .$

In the combination $\Phi .\rho ,$ we shall say that $\Phi$ is used as a prefactor, in the combination $\rho .\Phi ,$ as a postfactor.

110. If $\tau$ is any linear function of $\rho ,$ and for $\rho =i,\rho =j,\rho =k,$ the values of $\tau$ are respectively $\alpha ,\beta ,$ and $\gamma ,$ we may set

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

and also

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

Therefore, any linear function may be expressed by a dyadic as prefactor and also by a dyadic as postfactor.