14 
Proceedings of Royal Society of Edinburgh 
On a Rapidly Converging Series for the Extraction of 
the Square Root. By Thomas Muir, LL.D. 
(Received October 20. Read December 16, 1889.) 
It is well known that the square root of any commensurable 
number N is expressible in the form 
A+i 1 .... 1 1 
a x + a 2 + + a z + 2A + * * * * 
* * 
and that the (z + l) th convergent to this is 
K(A,fli,a 2 » • • • ♦ ,a z ) or ^ + K(a 2 , . . . , a z ) 
K(oq,a 2 , • • • • >&«) K(<q,& 2 , • • • 
where the continuant notation K( ) is explained by the example 
K(a v a 2 ,a 3 ) 
oq 1 
-1 a 2 1 
• “ 1 « 3 • 
Subtracting the ( z + l) th convergent from the 2 (z+ l) th we have 
K(A ,oq,fl 2 , . . . , a zy 2A y a^ y a 2y . . . y a z ) ^ K(A,cq,<% 2 , . . . ,a z ) 
K (a 1} a 2 , . . . ,a zy 2A,a v a 2 , ...,a z ) K (a v a 2 , . . . ,a s ) 
K(A,a 1 ,...,a z ,2A,a 1J ...,a z )K(a ly ...,a z )-K(a v ...,a z ,2A y a 1 ,... y a z )K(A,a ly ...,a z ) 
. . . y a z , 2A,nq, . . oq) 
= (-l) g K (a v ...,a z ) } 
K(a 15 . . ., a zy 2A y a ly . . . ,a z ) ) K(a v . . ., a z ) 
(-1 )' . 
K(a li ... J a z ,2A J a v ... i a z ) 
Similarly, on subtracting the 2(z + l) th convergent from the 4 (z+ l) th 
there is obtained 
(-l) 2g+1 
K(«i, • • • ,« z ,2A,a 1} . . . j a zy 2A y a ^ y . . . y a zy 2A y a^ y . . . , tt z ) * 
and so on generally, the numerators of the successive differences 
being evidently 
(-1)*, (-i r+\ (-i)^ 3 , (-i 
