ON DOUBLE POLYADICS — THE LINEAR MATRIX EQUATION. 395 



1 This symbolic method is due to H. B. Phillips: Some invariants and co- 

 variants of ternary collineations," American Journal of Mathematics, 36, 1914. 



2 Hamilton, Lectures on Quaternions, 1853, Cayley, A Memoir on the theory 

 of matrices, 1858. 



3 E. B. Wilson, On the theory of double products and strains in hyperspace. 

 Conn. Acad. Trans. 14, 1908. 



4 C. L. E. Moore and H. B. Phillips, The dyadics which occur in a point-space 

 of three dimensions, Proc. Amer. Acad, of Arts and Sci. 63, 1918. 



5 For the elementary theory, Bocher's Algebra may be consulted. 



6 See note 3. 



7 For the laws of p-way determinants see Amer. Journal of Math. 40, 1918, 

 by Lepine Hall Rice. 



8 Compare Joly's Appendix to Hamilton's Elements of Quaternions. 



9 This result is an extension of Hamilton's invariant property of his coeffi- 

 cients. 



10 Elements of Quaternions, 2nd Ed., Vol. I, Art. 348. 



11 In a series of papers over many years. They are all, I think, listed in the 

 bulletins of the Quaternion Association. See, in particular, Wien. Ber. 112, 

 1903, pp. 645, 1091, and 1533. 



12 The term "extent" is due to Sylvester. I have elsewhere given a sketch 

 of the present method in its relation to the work of Sylvester. Proc. Nat. 

 Acad, of Sci. 8, April, 1922. 



