14 



ALMAR NÆSS. 



M.-X. Kl. 



We then get by intuition that 



(5) o'ji = dj ■ e', = Ojk Elk 



which also, more exactly, can be found in the following wellknown way 



(6) (Xj = a ji e', = a ji Eik <tk 

 But as we also have 



a,- = ajk Zk 



we get 



a jiEik = ajk 

 which involves the following ;/- equations : 



(7) 



a ji f,i — aj^ 

 a'j, E,„ = ay 2 



/ 



a j i Eiu — Clin 



If we by Eij denote the cofactor of the element Sij in the determinant 

 of the es, these equations (7) give: 



a a = 



ajk Eik 



But as the f"s form an orthogonal matrix, we have: 



\Eij\ = I and E,k = Eik 



Therefore 



a ji = ajk Eik 



We now will form the space complement of the vectors a^ , (X.2 ■ . ■ . ^p 

 with respect to the new (primed) coordinate system. By definition it clearly is: 



Ci e.. 



.0) 



e^ e, . . 



^'11 ^'12 • 



a'p^ a p. 2 . 



e, fi , e, fo ' 



e,- fi , e, £., , 



ap i E-^ i ap i f., , 



e „ 



^i" 



(7 



1 " 



. a p„ 



C; E„ i 



Ci c, I i 

 «1 I E„ , 



ap i E„ i 



