Further Note on Factorizablc Continuants. 



193 



10. On glancing back at the preceding paragraphs it will be seen 

 that the new results obtained have arisen naturally out of an 

 investigation of the accidentally discovered identity (a) of § 2, 

 where the continuant is of such a form that it may be not inaptly 

 described as consisting of a diagonal of identical elements ensheathed 

 by the first 2n - 1 integers, and where the factors into which it is 

 resolvable form not one equidifferent series but two. 



Now there is another factorizable continuant, having some of 

 these characteristics, which is equally interesting in itself and 

 equally full of promise as a base for investigation, viz., 



(2tt-l)/3 



2n-l 



l.(/3 + 2;i-2) (2w~2)(/3+l) 



3-2% 



2{& + 2n-3) 

 5-2n 



2n-3 



3(/J + 2n— 4) 



7-2ra 



(w + l)()8+n-2) 



3 



(n-l)(i3+n) m(/3+m-1) 



-1 



= (o+/3) (a+ii+2) (a+/3+4) ( a +/3+2n-2). 



fX.) 



Here in the sheath enclosing the diagonal of as we have, as in the 

 previous case, the integers 1, 2, 3, ..., 2n-l going round the one 

 way, but we have also accompanying them as multipliers the 

 quantities /3, /3 + 1, /3 + 2, ..., /3 + 2>£-2 going round the reverse 

 way, and accompanying the latter as divisors the integers 2n - 1, 



2/t-3, , 1, -1, -3, , -(2/i-3). 



Using the fact that the right-hand side of (X.) is a function not 

 of a and /3 separately, but of a+/3, we find the corresponding 

 nil- factor continuant to be 



(. c 



2n - 1 

 1 ., _, 2n-2.. 



3^27^ s 2rc^3* 



2 .. 



5 - 2;i 



7-2^ 



™-l, 



-1 



71+1 



«. . n., 



y -I— <■ 



(XI.) 



