AI.MAK NTI-'-SS. i\I.-X. Kl. 



Ilic inntrix ol' the transformation M is therefore, by (11) where merely 

 compoiH Ilts in tiic ii'w coordinate system appear, the matrix ö ,^. 

 We notice that Mil also can l)c written: 



(121 "i/Jjk---gj,- 



the scalar pi-oduct hein.t;- in(le|)en(lent of the' cor)rdinate system. 



Assuming- the two systems are related to one another by the equations: 



(13) e/ = eijij 



i. e. : the components of c/ in the old system are: 



f' I I , f'io • • • ^"1 



But from (13) we get, remembering that C/ an e/ both are orthogoneil 

 systems of unit \'ectors : 



(14) e/-£y,e/ 



furthei": 



( 1 5 ) ir'j, = Ç\j ■ e/ = gjk f,/fr 



Hence, we get from (11) and (12): 



Cl, ■ fj = "ik/jk = ^'7/ = g/kfik = Clk • fjfik = Okifjieik 



(16) = akif;,kfii 



But as we here have to sum for both k and /, those two subscripts 

 may be interchanged. That is : 



(17) f^ikfjk ^ aiktnfik 

 This means : 



( 1 8) 0,-^ — aik en — Ujk fij 



or: 



(19) a,- = eijCij 



We now readily find the dyadic 0, expressed by the vectors of 0: 



(20) = e/ a,- = e/ ô,-^. ^^ = e/ ojk f,j ^/^ 



= e/ fij Ck a J A- = ey ex- a/k = 

 or, simply, by (19): 



= e/ a, = e/ fij aj = e^ a; = *. 



We thus see that is identical with 0. Ov, in other words : 

 If we denote a traiisfori)iatiou of the type T l>y means of a dyadic 0, 

 then /.s independent of any particiiUir coordinate system (or any normal 

 system of unit vectors) chosen, h'ith respect to which may be farmed or 

 written ont. 



