irrational functions . 345 
often possible, when the original function does not admit a 
rational expression, and can be effected sometimes directly, 
and sometimes by introducing a new variable. But it will 
first be necessary to reduce all irrational functions whatever 
to a definite form. 
Lemma. 
To reduce all irrational functions to a definite form. 
1. By successively multiplying numerators and denomina- 
tors into the same expressions, every irrational function may 
at last be reduced to a series of terms, whose numerators and 
denominators do not contain any fraction or negative index. 
jSll + and if (3 , y, a } b, c } are 
Thus. 
a. 
(, a + bc- n y n /■ • + b) 1 
functions involving fractions or negative indices, themselves, 
the reduction is continued in the same manner. 
2. Now multiply both the numerators and denominators 
of the expressions so reduced, by such multipliers, as will 
render the denominators rational. This factor is the product 
of all the different values of the denominator, with the excep- 
tion of the denominator itself. The new numerators will still 
consist of a series of terms not involving any fraction or 
negative index. 
3. If R, R, R, &c. denote functions of the form cx m + 
1 % 
C . X m “ l the irrational takes the form, 
R (x) R ( x ) 
2. V 
R (x) *L" R (*) 
a 
(p (x) b . c p (x) d . . . + 
R (x) 
4 
R (x) 
7T 
^ . . . -j- . 
4. By reducing the fractional indices of the factors to the 
l 
