158 Proceedings of Royal Society of Edinburgh. [sess. 
There is nothing to indicate that this is not viewed as a fresh 
discovery, notwithstanding the fact that Ramus’ paper of 1856 con- 
taining virtually the same identity was published in the same city. 
The other paper may he described as a careful study of finite 
continued fractions with the help of determinants. Instead of 
b Y , b 2 , . . . are used a 12 , a 23 , . . . ; and 
a p 
a p,p+i 
1 
a p + 1 
a p+l,p+2 
a q- 1 a q-l,q 
1 a q 
is denoted by 
K (p,q). 
Further, this determinant is spoken of as a “ Kjsedebrpksdeter- 
minant,” or, shortly, a “ K-Determinant ” ; and a section (§ 3, pp. 
149-152) is devoted to a statement of its properties. 
There is no need to rehearse all of these, the last portion (D) of 
the section being alone that which contains fresh matter. Opening 
with the double use of a previous property, viz., 
K(h,m) = K(h,k-l)-K(k,m) - a k _ 1>k K(h,Jc - 2)-K(k + 1 , to ) , 
K (h,n) = K(h,k- 1)-K(k,n) - a k _ hk K(h,k ~ 2)-K(& + 1, n) , 
where h, k , to, n are in ascending order of magnitude, the author 
eliminates 1) and obtains 
K(/i,m) K(k,m) I _ I K(k,m) K(k+l,m) 
K(/t,») K (k,n) ' a *~ lX (l ’ ; |K(fc,re) K(& + \,n) 
(a) 
Then by taking the particular case of this where k appears in 
place of h and k + 1 in place of k there results 
K (ft, to) K(& + 1,to) 
K (k,n) K(k+l,n) 
a k,k + 1 
K(& + 1,to) 
K(/£ + l,n) 
K(/r + 2 , to) 
K(& + 2 ,n) 
j 
which when applied to one of the determinants occurring in itself 
,to) K(&+1,to) 
t n) K(k+l,n) 
K(k + 2,m) 
K(k+2,n) 
K(& + 3, to) 
K(&+ 3,tz) 
gives 
