THE DYADICS WHICH OCCUR IN A POINT SPACE OF 



THREE DIMENSIONS. 



By C. L. E. Moore and H. B. Phillips. 



Received, November 2, 1917. 



In his Ausdehnungslehre, Grassmann gave a discussion of linear 

 transformations of space in which he considered each transformation 

 as determined by a Brucke,^ or fraction. By using products which 

 he called indeterminate,^ Gibbs showed that the transformations 

 could be determined by means of bilinear forms called dyadics. 

 These were applied to the study of linear vector functions in three 

 dimensions in the Gibbs-Wilson Vector Analysis. Extensions of this 

 to higher dimensions were given by Gibbs in his lectures on Multiple 

 Algebra an outline of which is contained in an article by E. B. Wilson.^ 

 H. B. Phillips* applied the dyadic to the study of projective transfor- 

 mations in a plane. The fraction of Grassmann does not lend itself 

 readily to algebraical manipulation. This is remedied by the dyadic 

 of Gibbs. The symbol used by Gibbs does not, however, suggest 

 the nature of the particular dyadic or the invariants and covariants 

 determined from it. In the paper of Phillips a symbolic notation 

 was introduced by which the dyadic appears as the product of a single 

 pair of letters from which invariants and covariants and combinations 

 with other dyadics are obtained by processes of multiplication analo- 

 gous to the Grassmann products of space elements. 



In this paper we have given an exposition of the symbolic notation 

 and have used it to discuss with some completeness the various dyadics 

 occurring in a point space of three dimensions. To aid in the under- 

 standing of this we first develop the elements of Grassmann's analysis 



1 Ausdehnungslehre, 1862, page 240. A good exposition of this is found in 

 Whitehead's Universal algebra, Chapter VI. Book IV. 



2 On Multiple Algebra, an address before the section of mathematics and 

 physics of the American Association for the Advancement of Science, by the 

 Vice-President. Proceedings of the American Association for the Advance- 

 ment of science, 35. This address is reprinted in the Scientific Papers, 2. 



3 On the theory of double products and strains in hyperspace. Transac- 

 tions of the Connecticut Academy of Arts and Sciences, 14, 1908. 



4 Some invariants and covariants of ternary collineations, American Journal 

 of Mathematics, 36, 1914. 



