1889 — 90 .] Dr T. Muir on Extraction of the Square Boot. 15 
and any denominator being got by taking 2A for its middle element, 
and writing the elements of the previous denominator both in front 
of this 2A and after it. 
Combining these results, we obtain the rather notable identity 
11 1_ 1 
+ a + a 2 + * * ’ -1- u z + 2A 4- 
* J * 
_ a [ K(a 2 ...,a z ) + 
(- 1) 2 
K(ofj,ci2) • • • j a z ) • • • j^z> %A,a v . . . ,a z ) 
..., a z1 2 A, a zf 2A, a^ 2 A, ..., a z ) 
1 
K 2 A,..., 2 A,..., 2A,..., 2A,..., 2A,..., 2A,..., 2A,. 
The rapidly converging character of the right-hand side is unmis- 
takable. There are, however, one or two transformations possible 
upon it which considerably enhance its value. 
In the first place, each denominator after K(a 1 , . . . , a z ) can be 
resolved into factors, viz., 
K(a 1} . . a z , 2 A, a li ...,a t ) = 2 K(a 1} . . ., a z )K{a v . . ., a z , A) , 
, 2A,..., 2A,..., 2A,..., a z ) = 2 K (a v ..., 2A,..., a g )K(a v ..., 2A,..., 
= 2 2 K(a 1} ...,a z ) K (a v ...,a z , A) , 
( xK(ffi 1 ,...,2A,...,fl 2) A), 
^Z5 E) , 
.,2A,...,2A,. 
2A,...,2A,...,2A,...,2A,..,2A 
= 2 3 K(a 1} . . ., a 2 )K(« x , ...,a zi A) 
x%,...,2A,...,a 2 , A) 
2A,..., 2A,..., 2 A,..., a z ,A) , 
So that if we put for shortness’ sake 
K 0 = K(a 1? . . ., a 2 ) , 
Ki = K(otj, . . ., a zi A) , 
K 2 = • • •} 2 A, . ., c£ z , A) , 
K 3 = K(a 1} . . ., 2A, . . ., 2 A, . . ., 2A, . . a z , A) , 
