364 HITCHCOCK. 



Hence the required scalar is twice the sum of two-row minors whose 

 main diagonals coincide with the main diagonal of the determinant 

 of <p. That is, (by the usual matrix theory), it equals 2m 2 , as we see 

 by developing (29). 



More generally, let Ai, A 2l - • -A p and Bi, B 2) - • • B p be any2p polya- 

 dics. Definition. We define the scalar 



((A l B u A.:B. 2 ,---,A p B p )) s (35) 



to be the sum of terms computed as follows: the leading term is the 

 product of A'-tuple dot products Ai : BiA 2 : B 2 - • A p : B p ; the other 

 terms are of like form, and are obtained from the leading term by 

 keeping the antecedents A\- • A p fixed in position while the conse- 

 quents Br • • B p are permuted in all ways; the sign of any term is 

 positive or negative according as the number of simple interchanges 

 needed to form the term is even or odd. 



If <p\, (p2, • • • , <p p are any p double polyadics, the scalar 



((<Pi, <&>' • •> <Pp))s ( 36 ) 



is defined to be the result of expanding each <p as in (23), multiplying 

 out, and adding the scalars formed from each term by the above defini- 

 tion. We may now suppose these p double polyadics to be all equal. 

 The scalar (36) becomes {{ap, <pr " to p factors))^ and may be abbre- 

 viated {(<p p ))s- It is now easy to see that 



((<P p )h=p!™ P (37) 



Proof. Expand <p as in (23). The terms of the product tp, <p,- • •, to 

 p factors are of the form 



EM U E y My, E,M,,--,E r M r (38) 



where subscripts may have any values from 1 to n; we see by (34) 

 that the scalar of this expression has for its leading term mum^jinkk 

 • • -virr, while, by the definition, the other terms are formed by inter- 

 changing the subscripts corresponding to the consequents. Therefore 

 the scalar obtained from each expression (38) is a determinant of 

 order p whose main diagonal lies on the main diagonal of the determi- 

 nant of tp, provided the p subscripts i, j, k, •■ ■ , r are all different; if 

 they are not all different the scalar of the expression vanishes because 

 it is a determinant with two or more identical rows as well as columns. 

 Each choice of subscripts of non-vanishing terms will occur p! times. 

 The sum of the scalars of all expressions (38) is then p! times the sum 



