366 HITCHCOCK. 



It need hardly be emphasized that formal analogies, when they exist, 

 are of much value, but do not in themselves imply more than certain 

 common laws of operation. 



6. The Scalars (36) are Sums of Cubic Determinants. 



The quantities denoted by (36) are more general than the coefficients 

 in the Hamilton-Cayley equation. When p has any value from 2 to n 

 such a scalar is a function of all the elements of each of the double 

 polyadics which enter. Suppose p = 2 and let <p\ and <#> be expanded 

 in the form (23). The scalar elements of <p\ and (p 2 may be designated 

 respectively by muj andm 2i] : We shall have in the indeterminate 

 product ipnp 2 pairs of terms of the form E ; Mh, E/M 2; + E ; Mi ; , E t M 2i 

 which, by forming the scalar by the definition (32), gives ww^?; — 

 mnfm2ji-\- tnij/niou — mijimnj- These four scalar terms are the same 

 as the cubic determinant 



mm, mm 



mm, mijj 



mm, m 2il 



(41) 



where the indices i andj are signant, 7 but the first index is non-signant 

 as might be expected because ((<pi, <p 2 ))s = {(<P2, <pi))s- We see that if 

 the respective determinants of <p\ and <p 2 be placed as non-signant 

 layers of a cubic matrix, the cubic determinant (41) has for its own 

 non-signant layers the two-row minors whose main diagonals occupy 

 corresponding positions on the main diagonals of <£>i and <p2. As a 

 verification, if <pi — ip 2 the cubic determinant does not vanish, but 

 becomes twice one of these minors, in agreement with the preceding 

 discussion of the coefficients in the Hamilton-Cayley equation for ip. 



Since i and j hare all pairs of values from 1 to n, it is clear that 

 ((<Pi<P2))s is the sum of all cubic determinants of the second order whose 

 main diagonals occupy corresponding positions on the main diagonals 

 of the determinants of <pi and (p- 2 . 



An analogous proof holds for all values of p up to n. Thus (36) can 

 be written as the sum of all the cubic determinants of order p whose 

 main diagonals are corresponding elements from the determinants of 



<P\, <f2,' " - , <Pp' 



In particular, when p = n, that, is, when we have any n double 

 polyadics, the scalar (36) is the cubic determinant of order n of which 

 the non-signant layers are formed by the square matrices muj, mm, 

 • • ' , m„n of these double polyadics. 



