irrational functions. 349 
apart, when the fluent of the whole cannot be found at once ; 
so conversely, the fluent of a series of terms may be found, 
when each separate term surpasses the powers of analysis. 
Thus we know 
tr'dq (x) 4- d <p (x) 4- • • /'V \ 
- y.J 
But we do not know 
d <p (x) 
?.(*) _+?(*) + •• 
d p (x) 
?(*) + $>(*) + • • • 
+ 
<p (*) + <p (x) + 77 ' " * 
( ,- * ( ' : 
Again, let <p (a:) be such a function of x> 
<4 
that <p 2 (x) = -7-; let cp (x) = x ; 
Then fdx . cp (x) == x . <p (x) 4* Jd cp (x) . X 
vi 
X . X 
Jon 8; ( :* 
^fd<p\x) . <p *(x) . * 
£ fd X , <p (x) 
j dx . <p (f) = * - f ^nflrn. 
' -s ' . ' , 
Which theorem admits farther extension, and may be kpplied 
to elliptic arches. 
j «j 1"! ; > 
Sfti O'... 
Ilia 
Should the above processes for rendering the fluxion 
rational fail us, we must attempt the fluxion at once in its irra- 
tional state, for which purpose I shall add a few miscellaneous 
observations. . -.T :vA 
to If (p (x), <p (x) be any algebraic functions, : v- 
then d\<p (x) -f» log? <p (x) } = dip (x) + — is an ailgc- 
8 a I >Mw 3 
braic expression? Wh&tte^er, th&fefoFej.rwe'>- mteeT with an 
