V 
irrational functions. 
347 
the series of terms above exhibited, may readily be found ; 
by introducing all the rational parts entirely tinder the radi- 
cals ; by reducing the indices of all the terms to a common 
denominator p ; by expanding all integer powers ; and by 
again reducing all the products and sums contained under — ’ 
to indices with a common denominator pi . These operations 
continued, will ultimately lead to the expression 
v- / yt / ~/\ tjx) R N 
S ^ S . S v » 9 where “ — - may be of any diffe- 
R (x) R l*> 
rent values in the different sums, but always of the form 
a. . os— I , 
ax + -r . . 
X 
bf + bz@~ l +... 
1 
R (x) 
q. itt* is said to be of ct (3 dimensions ; and if u — /3 be 
R (x) 
0 
dimensions of that rational part, whose dimensions are great- 
est ; then the dimensions of the whole irrational are — 4-4 — 
10 . The fluxion, and its dimensions in any irrational, may 
be found by applying this formula, d <p <p ... <p (x) j 
zzz D (p (p . . (p , D (p • . • D the D 
s a n a 3 » « 
only referring to the functional characteristic immediately 
succeeding it. 
Prop. VIII. 
To divide a fluxion into expressions admitting distinct ra- 
tionalities. 
Z z 
MDCCCXVI. 
