the Algorithm of Continued Fractions. 



299 



or into 



a 1 

 -I b 



or into 



c 1 

 -\ d\ 

 -1 e 1 

 ~1/ 



x b 1 



-Jcl 



1^1 

 -1 e 1 

 -1/ 



+ «x d 1 

 -1 <? 1 



c 1 

 -1 d 1 

 -1 e 1 



2. £. 



(abcdef) — (abc) . (jfcjf) + («6) . (cdef) 

 = (ab) . (cdef) + a . (foxfe/) 

 = a . {bcdef) 4- [cdef). 



Thus the whole of the properties of continued fractions are 

 deduced without algebraical calculation from a theorem which 

 itself springs immediately by inspection from the well-known 

 simple rule for the decomposition of determinants. 



If instead of a simple set a triple set of quantities be taken, as 



t| tg • • • H— 



! nic l ...m i 



w, n 2 . . .ni-i 



which, when t=sl, t-ss2, 1=8, £=4, &c. is to be interpreted to 

 hiean 



j»j; mj 



m, 



2> 



m-. 



8) 



— n 3 m 4 , 

 &c. respectively, the value of the determinant represented by 

 any such set being called T 8 -, we have in general 



(which, when m ? - and / { . ?i; are constant, becomes the characteristic 

 equation to an ordinary recurring series). The theorem corre- 

 sponding to the fundamental theorem for such triple sets will be 



l 2 •*• '*+»' 



m 1 m 2 ...m t+t / +1 )■ = * 

 n x n 2 ..♦ n i+i > 



m l m 2 ... nii 



n x n 2 



ni-i 



+ h X n t < 



n n 2 ...Wj_2 

 Lincoln's Inn, Sept. 15, 1853. 



H+\ 4+2 •••H + i' 



*{ m i+l m i+2 ...m i+ i> +1 

 n i+1 n i+2 ...n i+i > 



H+a ••• H+¥ 



tj tg ••• t{_ 2 



m l mc r ..m i - l y x < w^ +2 ... m i+t+1 



%i + 2 ... Wi + i' 



