﻿374 Mr. J.J. Sylvester on the Quadrature of the- Circle, 

 as follows, 



--1-4- 1 2 



i+ i + ir 20 



1 + -=- ad infinitum, 



the denominators of the partial fractions being all units, and the 

 denominators (after the first) the doubles of the natural series of 

 triangular numbers 1, 3, 6, 10 . . . . This is obviously the sim- 

 plest form of continued fraction for ir that can be given, and yet, 

 strange to say, has not, I believe, before been observed, Truly 

 wonders never cease ! 



At first sight it might seem as if the above-stated continued 

 fraction were incapable of teaching anything that cannot be got 

 direct out of the Wallisian representation itself that has become 

 transformed into it. Thus, ex. gr. y the convergent 



|,1 2 6 12 . 64 ' 



1+ iT IT iT T'.*'!>W 



2.2.4.4 

 is identical with the corresponding factorial product ' * * . 



But I think a substantial difference does arise in favour of the 

 continued fraction form, inasmuch as it indicates a certain obvious 

 correction to be applied in order that the convergence may be- 

 come more exact. For if we call 



w(n + l ) (ra + l)(n + 2) , . - .. 



_^ 1 i ^ . . . ad infinitum u n) 



1 + 1 + 



7t -4- n 

 we have u n =- . This shows that u n cannot remain finite 



1 + u n+ 1 



when n becomes infinite; for then u n+x would also be finite, and 

 consequently u n would be a finite fraction of infinity, which is a 

 contradiction in terms. 

 Hence ultimately 



n n . u n +i = n®-\-n, i. e. u n ^n, 

 or, in other words, 



1 1 1 



+ (»+l)-i+ (wt^-H 



. . . ad infinitum, 



converges (and, it may be shown, always in an ascending direction) 

 towards unity as its limit when n converges towards infinity. 

 Thus we may write when n is very great, 



