On an Extension of a Theorem in Continuants. 



361 



r 



5J i ^2 



OH c 2 r 



r 



— C| 















■c 3 ?» 



c 3 





 



h 



— c 4 



8 + *', 



which, lastly, is equivalent to 

 1 



7' 



6 2 



8+- 2 -^ 



?■ 



-1 





 



b 3 c 2 



8 + ^-c 3 r 

 r 







8 + 











b±c 3 



r 

 -1 





 

 



r 



Taking, therefore, continuants of the nth order, we have as a 

 general theorem. 



:K('8+ & - 1 - 



C^ 





5„c, 



8 + 



■CjJ* 



8+^-o 3 r 



'S+^-C-.r) 

 r / 



&n^n- 



(A) 



On putting d = c 2 = <?3= . . . =1 and writing /3 for 8—r, this 

 becomes 



r r 



h n - 



bn 



r / 



r J " 



h(«) 



which is the theorem given in the February Number, notwith- 

 standing the great apparent difference in the two ways of ex- 

 pressing it. In fact such theorems, as happens in the majority 

 of cases in determinants, are better expressed in words. Sta- 

 ting (a) in this way, we affirm that " If in any continuant the 

 ratio of the excess of any element of the minor diagonal over 

 the preceding element to the similar excess in the case of the 



