456 On a remarkable Modification of Sturm's 7%eorew. 



and if, by way of example, we take the fourth and fifth conver- 

 gents, these will be in the umbral notation represented by 



^9 ^8 K J h_K__^4_^ 



and 



«1 «2 «3 «4 ^\ «2 H «4 «5 



*1 h h h *1 *2 *3 *4 *6 



respectively. Hence 



Oa fla Aa «.«' 



N5D4-N4D5="2 "S "4 -6. X 



*2 ^3 ^4 *6 *2 *3 h ^1 



«2 % ^4 X ^2 ^3 ^* ^^ ^^ 



*2 *3 *4 *2 <^8 *4 ^5 ^1 



— "2 



flg ffg ^4 «, _ «2«3«4«5 «2 «3 «4 «1 ^ 



*2 *3 ^4 h K h ^4 *J h h h h h h K *1 



__ «2 f^3 «4 «5 X ^2 «3 «4 «1 



^2 *3 ^4 *1 *2 ^3 *4 *5 



__ «2 «3 «4 «5 X ^ ^';^' *^ " ^' ^*^ 



*i *2 h K K % '^4' 2?5 , ,. 



1 B 1 10 '^ i.tf4 6V+«¥ 



1 C 1 B 1 ^-e^-^^'iV 



i.e. = 1 D "" 1 C f;^-^^^"^^^»- 



1 1 I) 1 " ' 

 = 1x1 = 1, 

 as was to be proved. And the demonstration is evidently general 

 in its nature. We may treat a proper continued fraction irr pre- 

 cisely the same manner, substituting throughout \^ — 1 in place 

 of 1 in the generating matrix, and we shall thus, by the same 

 process as has been applied to improper continued fractions, 

 obtain 



I believe that the introduction of the method of determinants 

 into the algorithm of continued fractions cannot fail to have an 

 important bearing upon the future treatment and development 

 of the theory of Numbers*. — J. J. S. 



* If in the above matrix (M) we write throughout v — 1 in place of 1, we 

 have a representation of the numerators and denominators of the conver- 

 gents to a proper continued fraction, and such representation gives an im- 

 mediate and visible proof of the simple and elegant rule (not stated in the 



