go Mr. Woodhouse on the Independence of the 
necessary to state what I understand by the integral or fluent 
of an expression. 
V. Let <px denote a function of x; if x be increased by o> 
then <px becomes c p (x + of and <p (x + o), developed according 
to the powers of o, becomes cpx + Po + 
1.2 
2. 3 
o\ &c. 
where P is derived from <px, Q from P, R from Q, &c. by the 
same law ; so that the manner of deriving P being known, Q, 
R, &c. are known. The entire difference or increment of cpx 
is <p (x-f- 0) — <px; the differential or fluxion of cpx is only a 
part of the difference or P.o. If, instead of 0, dx , or x% be 
put, it is P . dx or P x * ; the integral or fluent 01 Px a is that 
function from which Px • is derived ; and, in order to re- 
mount to it, we must observe the manner or the operation 
by which it was deduced; and, by reversing such operation, the 
integral or fluent is obtained. Now, in taking the fluxion 
of certain functions of x, it appears there are conditions to 
which the indices of x without and under the vinculum are 
subject : hence, whether or not a proposed fluxion can have its 
fluent assigned, we must see if the fluxion has the necessary 
conditions. Expressions such as 
-, &c. have not 
i-\-x y VI— ^ 
these conditions; and consequently there is no function <px of x, 
such that the second term of the developement of <p(x + x-) is 
or, &c. There are, how- 
X' 
equal either to or — , or v __ r 
ever, integral equations front which such exptessions may be de- 
rived. Thus, let x — e*, then = x % let l + x = e* 
i-\-x 
%' 9 let x 
zV— 1 _ £ — zV—i 
. 2 V — I 
V 1. 
z=zZ* 
Now, from these equations, the differential equations x 
— l &c. may, by expunging the exponential 
V 
l—X 
