AI. MAK NÆSS. M.-\. Kl. 



Il iK)\v tilis ((Illation shall he written as a linear transformation of 

 (»idiiKify al/^cbia, V and v' iiiiisl he r(^'plac(j(] by two sets of scalar quan- 

 tities, vi/, their coiiipom nis in tin chosen coordinate .system. Let these 

 be V, and v,' such that \) C/ <', and 'o' - C,î'/, similarly the components 

 of a, being c/,/, j 1,2...//. Then (2) can be written : 



(3) 



"ll^'l + ''''12 ''2 + • • • + Oj^„V„ = v^ 

 n.^^i\ + a,,., v., + • • • + a.juVn =- f./ 



Oui "^'i + ^«2 î^2 + ■ ■ ■ + ^«« ^^" ~ ■^" 



Thus the niati-ix of this transformation, a,-/, expresses the same action 

 on Ü as does. If we put it, =^ (-'y^'V' ^^'^ ë^^ 



(4) =- e,e/rt,7 



and the coefficients of the dyadic therefore form the mati'ix of the trans- 

 formation which the dyadic carries out. 



We will here consider the dyadic as well as the matrix as merely 

 notations for the same geometrical "object", namely a transformation in 

 space. The latter can, of course, be realized as a "thing" which is inde- 

 pendent of any coordinate system chosen, can be defined without men- 

 tioning such a one, while in expressing the corresponding dyadic or matrix, 

 some sort of a coordinate system enters into both. Then the question 

 arises, whether or not those two notations (in the form given above, which 

 is the usual one) are independent of the coordinate system when the trans- 

 iormation, for which they shall be adequate expressions, is so. Here we will 

 try to show that the notion of a dyadic in such cases may have invariant 

 properties, which the matrix notation has not, and, accordingly, we cannot 

 very well say that every matrix is a dyadic and \-ice versa. 



Let there be given two sets of vectors (translations in space) 



f;. Ù, ... i> 



and C|i, CJ2, ... Cj„ 



They are supposed to be defined independent of any coordinate system- 

 Then there is in general one, and only one, (linear) transformation T car- 

 rying the vectors of the first set into those of the second set respectively. 

 And T, such defined, must be completely independent of any coordinate 

 system whatsoever. 



Let us choose some othogonal coordinate system e^, C.,, . . . e„ and 

 find the dyadic <P and the matrix M representing T with respect to this 

 system. We put: 



