ON DOUBLE POLYADK S — THE LINEAR MATRIX EQUATION. 357 



as one possible way of representing a four-dimensional assemblage of 

 elements on a flat surface. 



A polyadic is defined to be zero when and only when all its elements 

 are zero. Since our fundamental conception of a polyadic is as a sum 

 of polyads rather than as a mere aggregation of scalars, the equation 

 A = is to be thought of as equivalent to N K equations of the form 



aipb iq - -(ji s + (hjhq- -gis + + a hp b hq - -g hs = . (7) 



It is frequently useful to indicate such a set of equations symbolically 

 as ab • • g = 0, where, as before, the polyad is written symbolically for 

 the polyadic. 



2. POLYADICS AS VECTORS IN SPACE OF HlGHER DIMENSIONS. 



Investigations on polyadics may be distinguished according to 

 whether the A -dimensional character is important or not. If we agree 

 on some definite order among the fundamental polyads we may write 



E r = e p e q - e s (8) 



and let r run from 1 to N K while each of the K subscripts on the right 

 runs from 1 to N. With A r = A pq . . s we shall then have 



A = ^ 1 E 1 +^ 2 E 2 +-..+^ n E n (9) 



where, for convenience, n has been put for N K . 



The polyadic A thus takes the form of a vector in space of n dimen- 

 sions. This concept is justified if we introduce the multiple dot 

 product defined by 



(e a e 6 - -e ff ): (e p e g --e s ) = (e a «e p ) (e 6 -e 5 )- • -(e^e,,) (10) 



where the colon in every case indicates A'-tuple dot product. In 

 words 



Definition. The K -tuple dot product of two A-ads is the product 

 of dot products of corresponding factors. 



The /v-tuple dot product of two A'-adics is the sum of A'-tuple dot 

 products term by term. By virtue of the distributive law of multipli- 

 cation the relation (10) is symbolical of the iv-tuple dot multiplication 

 of two A-adics. When K = 2 we have Gibbs' double dot multiplica- 

 tion. 



By (8) and (10) it is evident that E, : E k is unity when i = k, other- 

 wise zero. Thus the fundamental polyads behave with respect to 

 multiple dot product as do unit vectors ei, e2, • • • e„ in space of n 





