Vol. 8, 1922 MATHEMATICS: F. L. HITCHCOCK 
83 
products, hence to dyadics whose antecedents and consequents are poly- 
adics, say double polyadics. The analogy may also be applied to certain 
other products. Thus the scalar ((ab, cd))^ or a.bc.d — a.dc.b may be 
regarded as a product of two dyadics M*N and dyadics found which are 
orthogonal with respect to star product, that is such that Mi*N;fe is unity 
when subscripts are equal, otherwise zero. We may similarly have star 
products for double dyadics and double polyadics. We shall have Hamil- 
ton-Cayley equations with respect to star products, where the scalars m 
are formed by star product just as those of this paper were formed by double 
dot product. The present paper is intended as introduction to more de- 
tailed treatment of some of these phases of multiple algebra. They have 
a bearing on ordinary algebra by virtue of the fact that a polynomial of 
degree K in N variables corresponds to a symmetrical polyadic of order 
K in space of dimensions.'^ 
1 In a series of Paris papers, C. R. Acad. Sci., 98, and 99 (1884) and elsewhere. The 
complete bibliography may be found in Bull. Inter. Assoc. Promoting Study Quaternions 
and Allied Branches, particularly for March, 1908. 
2 Paris, C. R. Acad of Sci., 99, p. 435. 
» "Note on the Linear Matrix Equation," Edinburgh Math. Soc. Proc, 22, 1904 (49-53). 
* The laws of signant and non-signant indices for ^-way determinants were given by 
Lepine Hall Rice, Amer. J. Math., 40, No. 3, July, 1918. 
'By forming the determinant of <p-\l. 
^Elements of Quaternions, Vol. I, 2nd. Ed., p. 498. 
^ Binary quadrics as vectors in 3-space, with what I here call star orthogonality, have 
been studied in great detail by Emil Waelsch. See the bibliographies referred to in 
Note 1. His method is given in Wien. Ber., 112, 1903, p. 645, 1091, and 1533, and called 
"Binare-analyse. " 
