Scientific Papers of J. Willard Gibbs Volume 2; Dynamics. Vector analysis and multiple algebra. Electromagnetic theory of light, etc

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1906 Excerpt: ...be straining things a little to call So-0 a vector. The combinations aX, fi/u., 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, $ 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 $ is identical with the study of ternary matrices. This expression of the matrix as a sum of products (which may be 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, aa $.'5r. Other important kinds of multiplication may be defined by the equations--(aX)J() = (ax/8)(xM), (aX):(/8M) = (a./8)(X.M). With these definitions J?!? will be the determinant of £, and will be the conjugate of the reciprocal of fr multiplied by twice the determinant. If $ represents the manner in which vectors are affected by a strain, will represent the manner in which surfaces are affected, and £ the manner in which volume3 are affected. Considerations of this kind do not attach themselves so naturally to the notation 0 = aSX+/JS/z + ySv, nor does the subject admit so free a development with this notation, principally because the symbol 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. VIII. QUATERNIONS AND THE AUSDEHNUNGSLEHRE. Nature, vol. Xliv. pp. 79-82, May 28, 1891. The year 1844 is memorable in the annals of mathematics on account of the first a...