1891-92.] Dr Muir on Convergents to the Boots of a Number, 17 
The next point to settle is, whether the convergents obtained in 
this manner, viz., 
1 n+1 3/z + 1 + Gtz- + 1 ^ 71 ^ + 1 0?^ + 1 
T’ hi + i ’ n^ + lOn + 5 ’ 
be not also obtainable in the ordinary way from some continued 
fraction. To do this a quite different mode of expressing the 
numerators and denominators of the convergents must be sought 
for. Utilising the facts that each denominator is the sum of the 
immediately preceding numerator and denominator, and that each 
numerator is the sum of the immediately preceding numerator and 
71 times the corresjDonding denominator, we readily see that the 
numerators are expressible in the form 
n 
1 
n 
n 
1 
n 
n 
n 
1 
-1 
1 
1 
-1 
1 
1 
- 1 
1 
n 
1 
. - 
1 
1 
j 
. - 
1 
1 
1 ' 
1 
1 
and the denominators in the form 
1 1 
1 n 1 
1 71 n 1 
-1 1 
j 
- 1 1 1 
-1 1 n 1 
. - 1 1 
j 
.-111 
. . - 1 1 
But, by subtracting the second row of each of these determinants 
from the first row, the third row from the second, and so on, each 
series is transformed into a series of continuants, viz., the first series 
into 
n+1 , 
n+1 n-1 
n+1 71-1 
-1 2 
j 
1 
H-J 
to 
1 
- 1 2 
and the second into 
2 
-1 
n-1 
2 
2 n-1 . 
- 1 2 n-1 
VOL. XIX. 
6/2/92 
- 1 2 
B 
