Vol.. 8, 1922 
MA THEM A TICS: h\ L. HITCHCOCK 
81 
Thus if = 2 we shall have 
((MiNi, MoNa))^ = Mi: N1M2: N2 - Mi: N2M2: Ni. 
If (pij <p2, ' • ' <pp are double dyadics defined as in (7) the scalar {{(pu 
<p2, . . . , (pp))s '^^ defined to be the result of expanding each (p as in (7), 
multiplying out term by term, and adding the scalars of all the terms of 
the product. 
If each M be expanded as in (6) the scalar elements of every (p will 
form a square matrix nits of order AT^ or w, so that each (p depends on 
scalar elements. The scalar ((^1, . . , (pp))s is the sum of cubic determi- 
nants of order p whose main diagonals lie on the main diagonals of the 
respective matrices ntts', the p matrices m^^ are to be regarded as p non- 
signant layers of a cubic matrix."* 
We may now suppose these p double dyadics to be all equal. The 
scalar just written becomes ((<^, ^, . . . to ^ factors)) 5 and may be abbrevi- 
ated {{<p'))s. 
Theorem. — The scalar {{<p^))s = pfntp where mp is the coefficient of 
<^"~^ in the Hamilton-Cayley equation for (p. 
Proof. Let (p be expanded as in (7). The terms of the indeterminate 
product {<py (p, ... to ^ factors) are of the form 
(E..M,-, E,.M,-, E,M„ . . . E,M,) (8) 
where any subscript will have any value from 1 to n. Since E^: M^ = mis 
the scalar of the expression (8) has for its leading term mamjjmkk - - • nirr 
and, by the definition, the scalar is the determinant of order p of which 
this is the leading term, a minor of the determinant of (p. Thus the scalars 
of all expressions (8) vanish except when the subscripts ]\ k, . . . , r are 
all diff'erent. Each choice of subscripts i, j, k, . . . , r will occur p! times. 
The scalar ((<^^)) $ is therefore p! times the sum of minors of order p taken 
along the main diagonal of the determinant of (p. By the usual matrix 
theory then {{(p^))s = pimp as was to be proved.^ 
5. Transformation Back to Matrices. — It has thus been shown how the 
invariants mi ... m„ of the Hamilton-Cayley equation for (p or 6 may be 
calculated by products which are closely analogous with Gibbs' products 
of ordinary dyadics. To complete the solution we need to see how these 
invariants may be formed directly from the given matrices Ai . . . 
and Bi . . . Bfi without the need of first forming (p by the rule (4). 
Consider first m^ the coefficient of li (p = SMN we have mi 
= 2M:N = Sab':a'b = Sa.a'b.b' = where As and Bs are the 
scalars of the dyadics corresponding to A and B, or, in matrix language, 
they are the respective sums of the elements of the main diagonals. Thus 
mi = A^sBis + ^25-^25 + . . . + AhsBhs- (9) 
For m2 we have, by the distributive character of all the steps involved 
in the process. 
