362 HITCHCOCK. 



5. The Hamilton-Cayley Equation for Double Polyadics. 



Let cp be a double A-adic expanded as in (23) so that the antecedents 

 are the fundamental A-ads Ei- • E n . Let^ be a second double A-adic 

 SBN and let its consequents first be expanded in terms of Er E n and 

 terms collected so that SI/ may take the form 



¥ - BiEi + B 2 E 2 + h B n E„ (27) 



so that the consequents are the fundamental A-ads. If we now form 

 the A-tuple dot product (f.^ it is clear that the scalar coefficient of 

 EiEt will be M; :Ba> or, in terms of the scalar elements of the polyadics, 

 will be ~Zmi s b s k summed on s. This agrees in form with the law of 

 multiplication of matrices of order n, a result which might have an- 

 ticipated; for if vectors are analogous to polyadics we should expect 

 matrices to behave like double polyadics, just as matrices of order N 

 behave like ordinary dyadics. The analogy holds of course only so 

 long as we are concerned with formal laws possessed in common by 

 the two algorisms. 



By virtue of this analogy, however, it is evident without further 

 proof that any double polyadic <p must satisfy an identity of the same 

 form as the Hamilton-Cayley equation belonging to a square matrix 

 of order n, namely 



<P 



"- ffl,^"- 1 + m,<p n -- V (-l) n mj = (28) 



where the coefficients my ■ -m n are scalars. The determinant of the 

 matrix or double A-adic <p-gl, where g is a scalar, is 



m n — g, ?»i2 , ?»i3, •••, '»ln 

 m-a. , w 2 2 — g, w 23 , •••, m- ln 



nin\ , w n2 , m n3 , ■■, m nn — g 



(29) 



and by writing <p in place of g and equating to zero we have the Hamil- 

 ton-Cayley equation. 2 



It is by no means necessary, however, to expand in terms of the 

 unit polyads in order to form the Hamilton-Cayley equation. Con- 

 sider for a moment the case A = 1, N — 3, when the double polyadic 

 becomes Gibbs' dyadic in ordinary space. The coefficient mi in the 

 Hamilton-Cayley equation becomes Gibbs' cps which, if the dyadic be 

 2ab is 2a- b, or symbolically wii = a-b. Again, the coefficient m- 2 



