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

Rh 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— where  or in quaternions  where  If we take the scalar product of the vector $$\Phi. \rho,$$ and another vector $$\sigma,$$ we obtain the scalar quantity or in quaternions  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