irrational functions. 353 
I shall conclude by observing that the fluxion may al- 
ways be made rational, if the fluent be wholly algebraic, or 
wholly logarithmic. Thus, if <p (a?) be any algebraic function, 
take oc = <p” 1 R ( if), 
Then d<p (os) =f^ d [<p<p~ x R (V)) = dR (??) 
and d (log.<p(®)) — d (iog. ^- 1 R(»)) = 
are both rational. If the fluent be of the mixed form 
<p (os) + log. (p (os'), its fluxion may be made rational, if R, R, 
s 1 
( ’ ' c- ! m 
can be so assumed that ip — 1 R (os) = q>T l R (a?) ; and it may 
i i 
always be effected by introducing two new variables. 
First let os R (y), and the fluxion becomes 
J R (z/)J 
^R (v) 4 - — — ; now let v = R"^ 1 <p(p-~ l R (%), an 
3 
d R (z) 
we get d R (v) + R *^ y which is wholly rational. 
\v i^klisbibirt orib vebcw Ids 
EDWARD FFRENCff BROMHEAD. 
, ; -• •/ .. > i 3 f 
1 ; ^ ' 3VBri e*w nod' 1 ’ 
Note.— As modern analysts have in general confounded the fluxions, either with 
the increments or the derived function s, it may not be superfluous to state precisely, 
~!j~ Kv \ ' J 
what is meant by the symbols d and D. 
I *13D\Q SOO. uO. spfftiit. *fQff HRr p 9m£8 9 uj QLPflA. 
If it be possible, (wflicn must Be shown in each particular case) to expand (p (x ~-f v) 
in the form (p (x) -j- <p (x) . -f <p (x) . v* + . . ; then <p (*■) is called the derived 
I a E 
function of <p (x), and rts v relation to <p (x) is thus expressed, <p (x) rr Dip (x). Hence, 
i 
if x be considered a function of itself, we have (x + v) ~ ( x ) + D (x) . v, and 
D (jr) r= i. 
. (T) M 
'M 
^1; n. 
%)M 
Now to avoid a constant reference to the variable x, of which other variables are 
considered as functions, we introduce fluxions. If y, x, w , . . . are functions of the 
