452 Transactions of the South African Philosophical Society. 



A' = ak - c 2 : *jj : 



- /- 16i - 2e/- ; 15 : 



5 





AA' - | aA - e 2 : *j? 



_ / 2 : lo : ) 2 _ 4^,2/2 

 / : 16 : j MJ 



i 15 ; 



: 16 



2 

 > 



- « 2 A 2 + e* : ^ 



2 : i A : 2 



+ f 4 ''■ 16 : " 2aAe * 



: 15 ; 



': 15 : . 



- 2aA/ 2 j \l j 





+^/'li!; : -i 



16; 

 16; 



_ 4.2/2 : 15 - 2 



4e/ ; 16;. 



On the other hand, the determinant on the right side of (IX.), if we 

 express it in a series of terms proceeding according to binary pro- 

 ducts of the elements of the first row and first column, is 



^f^k \} ( ®\ + ^ 2 + I }« 



; 156 ; 



- 2a 



; 16 



16 

 16 



74 + ; 15 

 1 + ; 15 



64 



A/ 2 - 2a 



15 ; 



15 



: ke< 



16 

 16 



15 >* 



15 ; e/ 



Now these two expressions, having five terms in common, will 

 manifestly be equal if we can show that 



A : 



; 156 



156 



15 



15 



16 

 16 



15 

 16 



and this is unnecessary because of the well-known theorem regard- 

 ing a minor of the adjugate. 



8. Before proceeding farther it is necessary to recall the series 

 of identities * 



\a I1 a 22 ...a lin \ — 



$11^22 I I ^11^23 



^11^32 I j ^11^33 



^I 1^/(2 I j ^11^/13 



Li j j (Jj 2 yi 



a 11 a 3n 



a ll a nn 



-f- a„ 







-f- I a ZI a 2 



n—3 



and to point out the effect of putting 



in the first, 



= a nl 



a. 



a, 



in the second, 



Trans. E. Soc, Edinburgh, xxix., pp. 47-54. 



