16 Proceedings of Eoyal Society of Edinbiirgh. [sess. 
contains the factor Jn and the first is rational. Consequently 
if we write 
0 + = 
and 
o+ir= 
2sjn 
0 + 1)" - o-if 
2Jn 
■ sjn + 
‘ Jn + 
2 
0 + 1)" + 0-1)" 
2 ’ 
we have the desired expressions for the numerators and denominators 
of the (2s~l)*’^ and convergents ; and all that remains to be 
shown is that for s = oc either convergent is equal to ^n. Taking 
the (2s- 1)^^ convergent, viz., 
0 + 1)"-^ -0 + 1 )^-^ 0 + 1 )^-^ + 0 - 1 )"-^ ^ 
2 ■ 2 > ’ 
we see that in order to prove that its limiting value i?, Jn v/e have 
merely to show that 
r o+i)""-o-i)^'^" _^ 
4 (,/« + !)" ■' + (>- if' 
X 
which is evidently the case ; and taking the 2s^^ convergent we have 
to establish the equally simple result 
s= X 
\Jn + 1 
o-i y 
V^ + ly 
= 1 : 
where the signs preceding the vanishing portions of the numerators 
and denominators bring out the fact that the convergents are alter- 
nately less and greater than 
