34$ Mr. Bromhead on the fluents of 
common denominator ( n ), the whole will consist of a series of 
R(x) 
terms 1 — . V 9 (*)* . <p (*)f . . . 
5, By expanding all the integer powers under the index 
— ; and again reducing the indices of the sums and products, 
which are under it, to a common denominator n' ; we shall 
by continuing the same operations, ultimately reduce the 
whole expression, to a series of terms of the form 
R (.r) 
a . r . ___ _ 
» <*> y s»'v/ s-y .. . 
m 
S denoting the sum of any number of terms such as follow 
it, wherein R (.r) may be different in each term, but always 
771 
of the form cxm + cx m —i 
I 
6. If every value of R (x) contains a factor (ax r + 
bx r — 1 + ...)”• "*• * * •> it may be taken entirely out of the 
radical ; and conversely the rational coefficient may be intro- 
duced entirely under the radical. 
7; When the surd is so reduced, that no rational factor 
can be withdrawn from the radical, it is said to be in its 
lowest terms ; and is said to be an irrational of the i 5f , y, 
or v th order, according to the number of the indices 
-7-1 — . Thus the general expression for a surd of the first 
n 
order 
R (*) 
is a series of terms, r-rV cx m + ex ™— 1 4 - • • * 
R O) 1 , 1 
3 
8. A more convenient general form for all irrationals, than 
