ON DOUBLE POLYADICS THE LINEAR MATRIX EQUATION. 389 



already noted under ?» 2 ; that is i, j, k — 1, 2,- •, h. Such a summa- 

 tion will hereafter be referred to as summation over the extent of the 

 equation, 12 and is implied in every symbolic equation. 



In general we may express m p symbolically by the equation 



p!m p = ((aib'i | a'ibi, a 2 b' 2 1 a' 2 b 2 , • • • , a p b' p | a,' p b p )) s (124) 



Expanding by the definition of Art. 5, the leading term is the product 

 of p factors of the form a r b' r : a' r b r , because the leading term is made 

 without interchanges among the consequents. These factors are the 

 same as a r -a' r b r -b' r and symbolically the same as A r sB r s- Thus if 

 we have a set of p subscripts i, j, h, ■ ■ ■ , r, s, t, the leading term in the 

 development of p!m p is 



AisBisAjsBjsAksBks- • • A r sB r sA s sB s sAtsBts (125) 



which is to be summed over the extent of the equation (11). 



The other terms in the expansion of (124) are obtained by making 

 all possible interchanges among the consequents, according to the 

 definition of Art. 5. Therefore so long as we maintain the polyadic 

 notation any term is a product of factors of the form a r b' r : a'sbs, 

 which is the same as a r -a' s b' r -b s . The vectors a r and a' s might 

 correspond to different matrices, likewise b' r and b s . Hence such a 

 factor cannot in general be translated into matrix notation if con- 

 sidered by itself. For it is of the essence of this transformation that 

 every vector a r be associated with its mate a' r and likewise every b r 

 with b' r ; then a r a' r is symbolically equivalent to A r and b r b' r to B r . 

 If, however, by a simple interchange of a pair of consequents in 

 the polyadic expression, we obtain a pair of factors (a r b' r : a' s b s ) 

 (a s b' s : a' r b r ), these may be developed as in (118) and (119) and yield 

 the two factors (A r A s )sXB r B s )s- Such factors occurred already in the 

 second term of w 2 and in the second, third, and fourth terms of m$. 



If three consequents change places among themselves, we obtain a 

 product of three factors of the form (a r b' r : a' s b s ) (a s b' s : a,' t b t ) 

 (a ( b' ( : a'rbr). It is important to notice that the order of subscripts in 

 the vectors a is the same as that in the vectors b but the order of accents is 

 reversed. It follows that when we develop after the manner of (122) 

 the vectors a yield the product of dot products a r - a'sa^a^a*, a' r , 

 while the vectors b yield the product bVbsbVb^b';, b r , where the 

 order of subscripts is the same, but the accents are on the first vector in 

 each dot product, instead of the second. These factors are equivalent 

 in matrix form to (A r A t A s )s and (B r B s B t )s, where the order of sub- 



