450 Transactions of the South African Philosophical Society. ' 

 neither will the left-hand member. We thus have 



2 



a cl f 





2 



d f 



a 



. b 2 c 2 





d 



. e 2 



d 



b 2 . A 





f 



e 2 



1 



c 2 A . 





a 



b 2 D 



a 



b 2 

 D 



2 f a d 



f . c 2 e 2 



a c 2 . F 



cl e 2 F . 



(VII.) 



This result could also have been got by considering the remaining 

 norms 



N(VA,Vn,VF)(&, d,e) 



N(x/A,x/D,x/F)(c, e, /) 



— a fact which is only natural when we note that these expressions 

 are obtained from the corresponding expression of §5 by performing 

 on it the cyclical substitutions. 



7. When zero elements exist in the first row beside the three 

 non-zero elements a distinction is necessary, as in §§3, 4, between 

 the case where the first place is filled by a zero element and the case 

 where it is not so filled. The latter case being the more closely 

 allied to that of §5 falls to be considered first. 



For convenience in writing let us limit the number of the said 

 zeros, A being 



a 



h 

 I 



m p 

 n q 

 o r 



e 



i J 



m n 



<1 

 .s 



t 



Then, as before, we have 



n<Wa + 0x/s + /n/u) = 



a 2 e 2 f 2 



a 2 . U S 



e 2 U . A 



/- ■ S A . 



But if we use 



r s 



to denote the minor of A got by deleting the 



rth, sth, ... rows and the r'th, s'th, ... columns, it is readily seen 

 that 



and 



S = a 



U = a 



\ 15 



; 15 



: 16 



; 16 



f 



- e- 



156 

 156 



156 

 156 



