Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/175

we adopt is a matter of minor consequence. In order to keep within the resources of an ordinary printing office, I have used a dot and a cross, which are abeady associated with multiplication, but are not needed for ordinary multiplication, which is best denoted by the simple juxtaposition of the factors. I have no especial predilection for these particular signs. The use of the dot is indeed liable to the objection that it interferes with its use as a separatrix, or instead of a parenthesis.

If, then, I have written $\alpha .\beta$ and $\alpha \times \beta$ for what is expressed in quaternions by $-{\text{S}}\alpha \beta$ and ${\text{V}}\alpha \beta ,$ and in like manner $\nabla .\omega$ and $\nabla \times \omega$ for $-{\text{S}}\nabla \omega$ and ${\text{V}}\nabla \omega$ in quaternions, it is because the natural development of a vector analysis seemed to lead logically to some such notations. But I think that I can show that these notations have some substantial advantages over the quatemionic in point of convenience.

Any linear vector function of a variable vector $\rho$ may be expressed in the form—

$\alpha \lambda .\rho +\beta \mu .\rho +\gamma \nu .\rho =(\alpha \lambda +\beta \mu +\gamma \nu ).\rho =-\Phi .\rho ,$

where

$\Phi =\alpha \lambda +\beta \mu +\gamma \nu ;$

or in quaternions

$-\alpha {\text{S}}\lambda \rho -\beta {\text{S}}\mu \rho -\gamma {\text{S}}\nu \rho =-(\alpha {\text{S}}\lambda +\beta {\text{S}}\mu +\gamma {\text{S}}\nu )\rho =-\phi \rho ,$

where

$\phi =\alpha {\text{S}}\lambda +\beta {\text{S}}\mu +\gamma {\text{S}}\nu .$

If we take the scalar product of the vector $\Phi .\rho ,$ and another vector $\sigma ,$ we obtain the scalar quantity

$\sigma .\Phi .\rho =\sigma .(\alpha \lambda +\beta \mu +\gamma \nu ).\rho ,$

or in quaternions

${\text{S}}\sigma \phi \rho ={\text{S}}\sigma (\alpha {\text{S}}\lambda +\beta {\text{S}}\mu +\gamma {\text{S}}\nu )\rho .$

This is a function of $\sigma$ and of $\rho ,$ and it is exactly the same kind of function of $\sigma$ that it is of $\rho ,$ a symmetry which is not so clearly exhibited in the quaternionic notation as in the other. Moreover, we can write $\sigma .\Phi$ for $\sigma .(\alpha \lambda +\beta \mu +\gamma \nu ).$ This represents a vector which is a function of $\sigma ,$ viz., the function conjugate to $\Phi .\sigma ;$ and $\sigma .\Phi .\rho$ may be regarded as the product of this vector and $\rho .$ This is not so clearly indicated in the quaternionic notation, where it would be straining things a little to call ${\text{S}}\sigma \phi$ a vector.

The combinations $\alpha \lambda ,\beta \mu ,$ etc., used above, are distributive with regard to each of the two vectors, and may be regarded as a kind of product. If we wish to express everything in terms of $i,j,$ and $k,$ $\Phi$ will appear as a sum of ${ii,ij,ik,ji,jj,jk,ki,kj,kk,}$ each with a numerical coefficient. These nine coefficients may be arranged in a square, and constitute a matrix; and the study of the properties of expressions like $\Phi$ is identical with the study of ternary matrices. This expression of the matrix as a sum of products (which may be