1889-90.] Dr T. Muir on Extraction of the Square Boot. 17 
As an example of the application of the series to the extraction of 
the square root, let us find by means of it ^300 or 10 J3. 
Since 
^300 = 17 + 
1 J. 1_ J_ 
3 + 8 + 3 + 34 + 
the series in this case is 
17 . K ( 8 > 3 ) _ 1 
K(3,8,3) 2K(3,8,3)K(3,8,3,17) 
1 
2 2 K(3,8,3)K(3,8,3,17)K(3,8,3,34,3,8,3,17) 
Now 
K(8,3) = 25, 
K(3,8,3) = 78, 
2K(3,8,3,17) = 2702, 
2K(3,8,3,34,3,8,3,17) = (2702) 2 - 2 
= 7300802 
so that the series becomes 
25 1 1 
17 + 78 78-2702 " 78-2702-7300802 “ ' 
Dividing 1 by 78, we obtain 
•0128205 128205 128205 128205 12...; 
Dividing this by 2702, we obtain 
•0000047 448233 976731 386057 81 ... , 
and dividing this by 7300802, we obtain 
•0000000 000006 499044 074436 12... 
All we have now got to do is to multiply the first of these quotients 
by add 17, and subtract the sum of the last two quotients. 
VOL. XVII. 
18/2/90 
B 
