Mr. BroriMad dti We fluents of 
Let (p (a*) be any irrational, ant? <p‘ (T) , <p (jr) &c. surds 
' ’■ 1 ■ - - ^ V. jnSun.erf* , \ 
deduced as in tlie Lemma. Then t , 
n dx . R x) f 'dx • R (x) 
fdx . t (x) =J~~ + j . f (*) + &C. 
where the fluent of the 1st term may always be found, and 
the other terms may often be rationalized by distinct substi- 
tutions, when we are unsuccessful with fdx. . <p (x). Again 
since in each of the terms, 
■ ; ) - ” r- - 4- : a) -v + ■{ ; 
fdx . R (*) R ( X ) 
S * a 
Jj “T(I) — * (®)» r^7 ) thay be reduced to a series of terms 
A 
of the form Ax n and ; therefore t^e fluent depends 
on a series of terms fdx . x n . <p (x), and fdx . (x J r a)~ n . <p (x). 
' ( f ) $ *1 % ' 1 % e y. ■ * 
In the latter case, the form of <p ( x ) is not changed by sub- 
I 
stating x for x 4- a, aritl the fluents of all irrationals are 
BJinli re. 
Cor. i. If we multiply the denominator of 
dx . R {x) , 
n f , Rfx) . dux 2, + fix 4- c + R(x) . \lciX 21 + 0x + y 
lornm 9riJ mrzsEmmiq evods * . 
by its rationalizing factor, the fluxion will be reduced to two 
terms jiwhieh admit distinct rationalities^ j/: f V/ . if ; ; 
Cor. 2. Sometimes by the substitution of a new variable, 
for some function of x, the fluxion will be divided into a series 
of terms, eacly of which may be separately made rational. 
Bu| unfort unat ely no general principle, has been discovered, 
f-j ) <\) 1 v ,J \ ' r ' - *'* / Y 1 r\ r 
to which these reductions can be referred. 
Cor.3. As the fluent of each terra can sometimes be found 
