18 ■ 
Proceedings of Royal Society of Edinlourgh. [sess. 
consequently, a connection with continued fractions is inevitable. 
For perfect agreement, however, with the convergents of a continued 
fraction, each continuant of the first series ought to be of a higher 
order than the corresponding continuant of the second series, it 
being necessary, in fact, that the latter should be the first principal 
minor of the former. This remaining obstacle is readily overcome 
on trial, the first series being further transformable into 
1 »-l 1 
1 72- - 1 
1 77 - 1 . 
-1 2 1, 
-1 2 n-l 
-1 2 77-1 . 
. -1 2 
5 
. - 1 2 77 -1 
. , -1 2 
whence it follows that the continued fraction 
1 + 
-1 
n-1 
furnishes exactly the same series of convergents as the new process. 
The fact that Jn is equal to this continued fraction is included in 
• the already known theorem 
— 7/i" 9 
n - 
'2m , 
2m + • • . . 
a particular case of which, viz., 7^=18, m = 4 dates back to 1613, 
seven years before the birth of Brounker.'"'^ 
It may be noted in passing that in obtaining, as we have done, 
a pair of unlike expressions for A and for B in the identity 
/- n — m^ 
sJn = m-\-—^ n 
2m +- 
(■n/tZ/ + 1)^ = A\/ 72/ + B , 
the equating of the members of each pair furnishes us with the 
evaluation of two continuants ; that is to say, we have 
1 n-\ 
-1 2 n-l 
. - 1 2 n-l 
-1 2 
(fn + ]y + (-iy(fn-iy 
r 
See Libri, Hist, des Sciences Math, en Italic, iv. pp. 87-98. 
