of certain differential Expressions , &c. 
2 49 
or 
-H 
b. i +'b 
b .'b (i+'£) (i+"b) 
+ 
2 . 2 * 2 2 . 2 : 
+ &c.} 
(9) 
in which, the last term, — ■ . 'P . 'Q . /. -^7-, is, in particular values of m. 
< m )b 
±.'P.'Q.l.±or A/P.'O./.-i-.or 
'P-Q-M- 
&C. 
each successive value being nearer the truth. 
Let6=x/|- v/C 1 — 6 2 ) or ^=v / 2'’ hence, the two formulas 
for the integral of dx V X | ) being equal, and the terms of 
the series b , '6, "6, &c. being respectively equal the terms of 
the series e, e\ e", e’", &c. we have P— P, 0 = 0 , and 
p 
- b{ 
-•C 1 -*Q) 
2.2 >2.2. 2. 2 1 J I 2” 1 -<• ("V& 
The two forms (5) (7) , are fully adequate to the computation of the 
integral of dx^/l^ ■ in all values of the series (5), in- 
volving £, e', e" , e"\ &c. is to be used, when e is any value between 
o and s/\ ', and the series (7), involving b .'6, "6, '"6, &c. is to be 
used, when e is any value between and 1, or, what is the 
same thing, when b is any value between o and s/ ■§. 
From the preceding forms may be deduced a very curious and 
remarkable theorem for the circumference of a circle, which I 
shall now exhibit. 
By former substitution, u'== a: v / / ( ; 
and when x= / {l +~y u ’= *■ 
Hence,y v(I _ il . a)(I _ ea ^ ) {x — V(l+6) ) — t /u' 0— *)> 
(* =1 ) = ^ - 2 /tt (* - 1 )- 
Kk 2 
