Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/71

as is at once evident, if we suppose $\Phi$ to be expanded in terms of $ii,ij,$ etc.

116. Def.—The (direct) product of two dyads (indicated by a dot) is the dyad formed of the first and last of the four factors, multiplied by the direct product of the second and third. That is,

$\{\alpha \beta \}.\{\gamma \delta \}=\alpha \beta .\gamma \delta =\beta .\gamma \,\alpha \delta .$

The (direct) product of two dyadics is the sum of all the products formed by prefixing a term of the first dyadic to a term of the second. Since the direct product of one dyadic with another is a dyadic, it may be multiplied in the same way by a third, and so on indefinitely. This kind of multiplication is evidently associative, as well as distributive. The same is true of the direct product of a series of factors of which the first and the last are either dyadics or vectors, and the other factors are dya.dics. Thus the values of the expressions

$\alpha .\Phi .\Theta .\Psi .\beta ,$⁠$\alpha .\Phi .\Theta ,$⁠$\Phi .\Theta .\Psi .\beta ,$⁠$\Phi .\Theta .\Psi$

will not be affected by any insertion of parentheses. But this kind of multiplication is not commutative, except in the case of the direct product of two vectors.

117. Def.—The expressions $\Phi \times \rho$ and $\rho \times \Phi$ represent dyadics which we shall call the skew products of $\Phi$ and $\rho .$ If

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

these skew products are defined by the equations

${\begin{aligned}\Phi \times \rho &=\alpha \,\lambda \times \rho +\beta \,\mu \times \rho +{\text{etc.,}}\\\rho \times \Phi &=\rho \times \alpha \,\lambda +\rho \times \beta \,\mu +{\text{etc.,}}\end{aligned}}$

It is evident that

$\{\rho \times \Phi \}.\Psi$	$=\rho$	⁠	$\Psi .\{\Phi \times \rho \}$	$=\{\Psi .\Phi \}\times \rho ,$
$\{\rho \times \Phi \}.\alpha$	$=\rho \times [\Phi .\alpha ],$		$\alpha .\{\Phi \times \rho \}$	$=[\alpha .\Phi ]\times \rho ,$
$\{\rho \times \Phi \}\times \alpha$	$=\rho \times \{\Phi \times \alpha \}.$

We may therefore write without ambiguity

$\rho \times \Phi .\Psi ,$⁠$\Psi .\Phi \times \rho ,$⁠$\rho \times \Phi .\alpha ,$⁠$\alpha .\Phi \times \rho ,$⁠$\rho \times \Phi .\alpha .$

This may be expressed a little more generally by saying that the associative principle enunciated in No. 116 may be extended to cases in which the initial or final vectors are connected with the other factors by the sign of skew multiplication.

Moreover,

$\alpha .\rho \times \Phi =[\alpha \times \rho ].\Phi$⁠and⁠$\Phi \times \rho .\alpha =\Phi .[\rho \times \alpha ].$

These expressions evidently represent vectors. So

$\Psi .\{\rho \times \Phi \}=\{\Psi \times \rho \}.\Phi .$

These expressions represent dyadics. The braces cannot be omitted without ambiguity.