Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/40

28. This principle may be used in the transformation of more complex products. It will be observed that its application will always simultaneously eliminate, or introduce, two signs of skew multiplication.

The student will easily prove the following identical equations, which, although of considerable importance, are here given principally as exercises in the application of the preceding formulæ.

29. $\alpha \times [\beta \times \gamma ]+\beta \times [\gamma \times \alpha ]+\gamma \times [\alpha \times \beta ]=0.$.

31.	$[\alpha \times \beta ].[\gamma \times \delta ]$	$=(\alpha .\gamma )(\beta .\delta )-(\alpha .\delta )(\beta .\gamma ).$
32.	$[\alpha \times \beta ]\times [\gamma \times \delta ]$	$=(\alpha .\gamma \times \delta )\beta -(\beta .\gamma \times \delta )\alpha =(\alpha .\beta \times \delta )\gamma -(\alpha .\beta \times \gamma )\delta .$
32.	$\alpha \times [\beta \times [\gamma \times \delta ]]$	$=(\alpha .\gamma \times \delta )\beta -(\alpha .\beta )\gamma \times \delta =(\beta .\delta )\alpha \times \gamma -(\beta .\gamma )\alpha \times \delta .$
33.	$[\alpha \times \beta ].[\gamma \times \delta ]\times [\epsilon \times \zeta ]$	$=(\alpha .\beta \times \delta )(\gamma .\epsilon \times \zeta )-(\alpha .\beta \times \gamma )(\delta .\epsilon \times \zeta )$
		$=(\alpha .\beta \times \epsilon )(\zeta .\gamma \times \delta )-(\alpha .\beta \times \zeta )(\epsilon .\gamma \times \delta )$
		$=(\gamma .\delta \times \alpha )(\beta .\epsilon \times \zeta )-(\gamma .\delta \times \beta )(\alpha .\epsilon \times \zeta ).$
34.	$[\alpha \times \beta ].[\gamma \times \delta ]\times [\gamma \times \alpha ]$	$=(\alpha .\beta \times \gamma )^{2}.$

35. The student will also easily convince himself that a product formed of any number of letters (representing vectors) combined in any possible way by scalar, direct, and skew multiplications may be reduced by the principles of Nos. 24 and 27 to a sum of products, each of which consists of scalar factors of the forms $\alpha .\beta$ and $\alpha .\beta \times \gamma$, with a single vector factor of the form $\alpha$ or $\alpha \times \beta$, when the original product is a vector.

36. Elimination of scalars from vector equations.—It has already been observed that the elimination of vectors from equations of the form

$a\alpha +b\beta +c\gamma +d\delta +{\text{etc.}}=0$

is performed by the same rule as the eliminations of ordinary algebra. (See No. 9.) But the elimination of scalars from such equations is at least formally different. Since a single vector equation is the equivalent of three scalar equations, we must be able to deduce from such an equation a scalar equation from which two of the scalars which appear in the original vector equation have been eliminated. We shall see how this may be done, if we consider the scalar equation

$a\alpha .\lambda +b\beta .\lambda +c\gamma .\lambda +d\delta .\gamma +{\text{etc.}}=0,$

which is derived from the above vector equation by direct multiplication by a vector $\lambda$. We may regard the original equation as the equivalent of the three scalar equations obtained by substituting for $\alpha ,\beta ,\gamma ,\delta$, etc., their ${\text{X-}},{\text{Y-}}$, and ${\text{Z-}}$components. The second equation would be derived from these by multiplying them respectively by