8 Al. MAR NÆSS. M.-N. Kl. 



(9) A ^ c, ^7,1 Cl f c,a,\^ e.., + . . . . + e, a,„e« 



Let us here iiUrrxluce a vector system x, deuned Ijy 



( 10) xi = ÇjOj,- 



Then we have 



(11) A ^y., C. 



The system x,- is said to be coi/jiiga/c to the system a,. Two conjugate 

 systems of vectors are determined by the rows and columns of the same 

 square matrix.* The dyadic A,, the conjugate to A, is then the following: 



(12) Ac = erx,. 



In an analogous way the definition of triadics, tetradics .... polyadics 

 is extended to S,i. A triadic, or tensor of the third order, is any sum of 

 the form : 



< 1 3) e,- ey e/t rt,/x- , ,;>, k = i,2. . . . . u 



or any quantity, which can be broken up into terms of this kind, and thus 

 wholly determined by a cubic matrix a,jk. If we instead of e, C; C/t have 

 the indeterminate product of p unit vectors multiplied by a scalar, /. e. : 



(14) e/ie,o . . . C^n,^,., . . . ,^ 



we get an elementary pol3-ad of the p"' order, and an\- sum of such 

 quantities is called a polyadic (or tensor) of the p"' order. As above, the 

 ;/ scalars a,^ ,., . . . ,. suffice for the determination of the polyadic, which 

 is called complete when these ;/ scalars are independent of one another. 



The special dyadic which transforms any vector into itself is called 

 the idemf actor (EinheitsdN-ade) and denoted by /. It is always reducible to 

 the form 



(15) / = e, e, (sum for /,) 



which follows immediate!}' from the fact that the corresponding matrix of 

 transformation in this case must be the unit matrix. 



The scalar (dot) product of two dyadics, which is frequently used in 

 the following, is defined in S„ exacdy in the same way as in S^.** It may 

 be expanded, according to the distributive law of multiplication, into a sum 



Concerning the properties of conjugate vector systems in three-space, see Zur Theorie 

 der Triaden von Almar Næss (Kristiania 1921I. 

 " See GiBBS- Wilson : Vector Analysis, p. 276. 



