372 Transactions of the South African Fhilosoiohical Society. 



we have the important proposition that a necessary and sufficient 

 condition for the equaHty of the n fractions with which we started is 

 the vanishing of the n-1 determinants obtainable from N,, N3, ..., N„ 

 by putting 



in place of 



ttj, a2, ••., ^rt—i, <x„, tiJ-^—i 



tto, ttj, ..., a^i^2i ^n—n ^rv 



(IX) 



Further, if we call those new determinants M,, M3, ..., M„, we have 

 as an analogue to (V.) 



-a,Q2 = M^ — aM2, 

 a\Q^ = M4 - aM, + a,a^l£,, (x) 



3. Instead of taking as our initial condition the vanishing of 

 |N,D,|, |N,D3|, ..., |N,D,J 

 we might with equal propriety have taken the vanishing of 



|N,D,|, |N,D3|, IN3DJ, ..., |N,„D,.!-* 

 The proposition then corresponding to (IV.) is 



iNAl = |" 



e, 





N3l>.l = ! 





a^ 



a. 



e, 



a. 



a^ 



e. 



a, 



a. 



. 



a. 



a. 



e. 



a. 



a, 



e. 



a. 



a?. 



a^ 

 a, 

 9, a, a. a. 



(XI) 



from which we see that the co-factor of - ag in | N1D2 1, the co-factor 

 of - al in \ N2D3 1, the co-factor of - al in | N3D4 1, are the deter- 



* It may be worth noting that these are at once transformable into 





a. 



«! 



a^ 



tto 



- 



&i 

 6. 





+ 



„2 ^2 «I 









&x 



tto 



. 





a. 



tto 







&x 



tto . 



2 



?)3 



«! 



ao 



h. 



ttl 



aa 





cu 



fl, 



«o 



_ 



h. 



«i tto 



+ af, 



^ 



a,. 



«i 



a^ 



Jh 



6r 





a. 



a^ 



«I 





h 



^2 rti 





^3 



63 



&. , 



