of certain differential Expressions , &c. 
247 
&c. = 'Q ; then, since 
, b , b .'b , b.'brb 
and T + — + Tzrrr 
nearly = = i d (W (l+C) ) - * 
d _£_ _ ft*- 'ft -"ft g ■ (i+'ft) ( !+''*>) -• (i+g) 
2.'P 2.2.2 . 2.2. 2 * 
we have 
i-f ft* tP \ 
vt.i+r) (1+r^ 1 ) 
i+w a / 
&*• ^ . ft*-ft -'(t+^) (i+ w ft) 
2.2 ' 2 . 2 . 2 . 2 
(7) 
ftVft. w ft (1+ V ft) (i+ >v ft) (I+ W ft) . 
b . 'b "ft 
+ &C. }h.iog.f+ \/(i+D; 
f= v \/{ 
-{ 
+ &C.} 
+ £^{i + ii 2 . + 
I 2 ” L 2 * 2.21 
since the last term, to wit, 
£-j-v/(i +£*)> or , may be neglected, on account of 
its smallness. 
Suppose it were required to find, from this form, the whole in- 
tegral of dx l " ' {Zl x i ) fr° m x = o to x = 1, put v = -fy, then 
ft 1 .'ft ."ft 0.(1 +'ft) ( 1 +"ft) ( 1 +) 5 ) 
similarly, 2"=^, 2"'= &c. 
Consequently, since 
f= v s/ Y 
(since 6 2 
4 'ft 
(i+'ftf I >/ 'ft 
l 1 +xP 
l+'ft 
2 \ 2 1 2 . 2 . 
'ft. (l+'ft) (i+"ft) „„ , 'ft ."ft . (I + v ft) d+''ft) (i+'"ft) 
+ &c. 
-f & ‘ --jr- - (?+v / ( 1 +D)» denoting the Naperean loga- 
rithm,) when v — 
/=- 1 
ft /ft.I+'ft I , ft'ft (l+'ft)(X+"ft) I 1 Qr^» 1 
~~ 2 l r v'ft 2.2 . r • v w ft 1 
MDCCCIV. 
i + ^(i+i 3 )' 
V/3 
Kk 
