1922. No. 13. ON A SPECIAL POLYADIC OF ORDER // — p. g 



of products of dyads, this sum being, of course, independent of the particular 

 form in which the dyadics are written. Let the dyadics be for example: 



(, 1 6 ) .4 = e, a, and B = e, hi. 



Hence the product, which is also a dyadic, may be written 



(17) AB = ç,a,B 



and the vector system defining this new dyadic (/. c. the i"' vector of the 

 system, / running from 1 to /;) is: 



(18) a,- B ^ a, • e^ by = ^o ^J 



or, by being equal to Qkàjk: 



(18') Ci, ■ B = Ç/c a,j bj k, sum for y and k 



Let us by x/' denote the vector system which is conjugate to the b's 

 (i. e. a s3-stem such that its /''' vector has its components in the /'''' column 

 of the matrix è,j defining the dyadic B). That is: 



(19) y.!' = tjbji. 

 Therefore : 



(20) a,- • 5 = Zk Oij hjk = e/c a,- • yJ' = a,- • >;/ e^ 



a result which is obtained directly' by observing that : 



(21)' B = Zk\ik = yJzk 



and, accordingly : 



(22) a, • B = a,- • \ykf' Zk\ = (a, • yJ) Zk = Zk a, • yJ 



This only means that if c,y is the matrix of the dyadic .4 • B, then 



(23) Cij = a,- yj' 



As, for any vector V, .4 • Z? • Ï» = .4 • (i? • ïi) is the resulting vector when 

 B and A acting in succession upon the vector tJ, this simply contains the 

 multiplication law of two matrices, which, hence, is compatible with the 

 law of (scalar) multiplication of two dyadics. 



§ 3. Remarks concerning the vector product and the reciprocal 



vector system. 



As is well known, the vector product of two vectors a and b, denoted 

 by a X b, in three-space is a vector whose components are the two rowed 

 determinants which can be formed from the matrix of the components of 

 the factors, i. e. from the matrix : 



