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

160 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— 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.