Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/176

extended to matrices of any order) affords a point of departure from which the properties of matrices may be deduced with the utmost facility. The ordinary matricular product is expressed by a dot, as $\Phi .\Psi .$. Other important kinds of multiplication may be defined by the equations—

$(\alpha \lambda )_{\times }^{\times }(\beta \mu )=(\alpha \times \beta )(\lambda \times \mu ),$⁠$(\alpha \lambda ):(\beta \mu )=(\alpha .\beta )(\lambda .\mu ).$

With these definitions ${\tfrac {1}{6}}\Phi _{\times }^{\times }\Phi :\Phi$ will be the determinant of $\Phi ,$ and $\Phi _{\times }^{\times }\Phi$ will be the conjugate of the reciprocal of $\Phi$ multiplied by twice the determinant. If $\Phi$ represents the manner in which vectors are affected by a strain, ${\tfrac {1}{2}}\Phi _{\times }^{\times }\Phi$ will represent the manner in which surfaces are affected, and ${\tfrac {1}{6}}\Phi _{\times }^{\times }\Phi :\Phi$ the manner in which volumes are affected. Considerations of this kind do not attach themselves so naturally to the notation $\phi =\alpha {\text{S}}\lambda +\beta {\text{S}}\mu +\gamma {\text{S}}\nu ,$ nor does the subject admit so free a development with this notation, principally because the symbol ${\text{S}}$ refers to a special use of the matrix, and is very much in the way when we want to apply the matrix to other uses, or to subject it to various operations.