1903-4.] Dr Muir on the Theory of Continuants. 
157 
from the expression on the left merely in having k + 1 in place of 
k ; so that there results 
— = ( - l) 2 ^^+l(N A+2 ,?iN fc+3jS — Nfc +2>s hl k+3,n) 
= (-iy 
a,, a. 
^'+l-N"s+3,?i 
This also, it will be seen, is connected with a result of Schlafli’s ; 
for putting s = n - 1 we have* 
1 N 7c+m _i 
N; 
1 -Nfc+l,»l 
N, 
— ( — l) n k a k a k+1 
. . a 
n 9 
which becomes identical with Schlafli’s last proposition on trans- 
posing the two rows of the determinant and (what is equally im- 
material) putting k= 1. 
Thiele, T. N. (1869, 1870). 
[Bemgerkninger om IvjsedebrOker. Tidsskrift for Math. (2), 
v. pp. 144-146. 
Den endelige Kjsedebrpksfunktions Theori. Tidsskri forft 
Math. (2), vi. pp. 145-170.] 
The first of the two notes comprising Thiele’s first paper con- 
tains only one result, viz., 
&i , 
a, + - + \ 
1 a 9 — 
2 a 0 
(a 1} a 2 , ... , a n ) 
(a 2 , . . . , a n ) ’ 
where (a v a 2 , ... , a n ) is used to stand for 
a x b l 
- 1 a 2 b 2 ... 
®»-i K^l 
- 1 a n 
* In giving to N s+ i )S , N s+ 2 ,s, N S+3)S the values 1,1,0 which are necessi- 
tated by assuming the generality of the recursion-formula 
= Nfc+I,n + <%N4-|-2 ,w 
Worpitzky forgets to note that in these cases the proposition N^ n = N n ,/ c , used 
by him in the demonstration, does not hold. 
