1922. No. 12. AX INVARIANT PROPERTY OF THE NOTION OF A DYAIMC. 5 



F^urther let the components of f,, (^,, and vi/ (for any /I in this system 

 be respectively : 



./'1' Ji-i> • • • J'» 

 g'l> g"2' ■ • • g", 



ai\, ai.2, . . . a,„ 

 whereby : 



Then we have for all j's i. e.: J = 1,2, .. i/ 

 (5) <P ■ fj = gy 



or (6) e,a/- f/ = ç,gj, 



or (71 iV- f; =-^y,- 



or (81 ^'Kfj^ ~ gj' 



giving in all ;/- equations for the »^ unknowns a,k, which thus can be 

 determined according to methods well-known in ordinary algebra ^ Hereby 

 is as well as M determined, as M is nothing but the matrix a,k- 



Let us now introduce another (orthogonal) coordinate system e^ , C, , 

 . . . e«'. We will find the dyadic <P and the matrix M denoting the same 

 transformation of space, T, and formed with respect to this new coordi- 

 nate system. 



Then ø must be of the form : 



(9) fp = e/ a, 



Let the components of a, in the old system be 



a, J, Ö,.,, . . . a,,, 

 and in the new svstem: 



Further let the components of f, and Cj, in the new system be primed: 



and g'ii, gio, . ■ . g in 



respectively. 



Then we have, similar to (7): 



(10) C^'-fj = g'j' 

 or ( 1 1) "ii/jk = g J, 



» The solution is ver„v easily carried out by means of vector analysis operations, see 

 On a Special Polyadic of Order n—/> etc. by Ai.mar Næss § 14. (Videnskapsselskapets 

 Skrifter. I. Mat.-naturv. Klasse 1922. No. 14.I Kristiania 1922. 



