1891-92.] Dr Muir on Gonvergents to the Roots of a Numher. 15 
Note on a Theorem regarding a Series of Gonvergents 
to the Roots of a Number. By Thomas Muir, LL.D. 
(Read January 4, 1892.) 
If the positive integral powers of f n + 1 be taken, and the 
expansion of each be separated into two parts, rational and irrational, 
thus — 
0 + 1)1 ^ 
0 + 1 )' = 
0 + 1 )^ = 
0 + 1)1 ^ 
0+1)5 
Jn + 
2 v/?^ + 
{n + 3) Jn + 
{\n + 4) J 71 + 
{ii^ \0n b) J 71 + 
1 , 
{n + 1 ), 
(3?^+l), 
{ri^ + 6/i + 1), 
(5?^2+10^^+l), 
then the ratio of the rational portion to tlie coefficient of Jn in the 
other portion is approximately equal to fifi, the convergence being 
perfect when the power of the binomial is infinite. This is the 
simplest case of a theorem discovered by the late Dr Sang, and 
enunciated by him as the result of a process of induction in his 
paper “ On the Extension of Brouncker’s Method to the Comparison 
of several Magnitudes” {Proc. Roy. Soc. Edin., vol. xviii. p. 341, 
1890-91). 
It seems desirable to have some further investigation into this 
curious proposition, and to try if possible to bring it into relation- 
ship with the already known mode of obtaining convergents to 
sJti, or, failing this, to show that the two modes are perfectly 
independent. 
In the first place, a proof of the theorem is wanted. To obtain 
this we must get a suitable expression for the convergent, that 
is to say, for the rational portion and the coefficient of Jn in the 
irrational portion of the expansion of {Jn + l)h Now manifestly 
2 ^ 
and in this when is odd the first fraction contains as a factor 
and the other is rational, while when r is even the second fraction 
