35^ Mr. Bromhead on the fluents of 
I'doc . j<p (a?) + (a?) | j (a?), and we have 
J'dx . <p (a?) = — Jdte . ft (.a?). 
If therefore n (a?) be a simpler expression than <p (a?), the 
fluxion will be reduced to a simpler form. In order to find 
7 r (a?) ; <p (a?) is assumed with indeterminate coefficients, so 
s 
■ ! . ; • _ A / 
■ ■■ . , - . . ■, •. _ . j f \ \ 
that its fluxion may be of the same form as dec . <p (•*■)• Now 
equate the similar terms in the two expressions, and the inde- 
terminates may be found. But as there may be more equa- 
tions than indeterminates, we add tt (a?) a function of the 
same form, and containing indeterminates of sufficient num- 
ber, to satisfy all the deficient equations. We shall thus have 
D <p (a?) = (p (x) + 7r (a?) and <p (i) jdx. j <p (a?) + (a?) } 
by which the difficulty may be reduced to findin gfdx . tt (a?). 
Reductions of particular kinds were discovered by Simpson 
and others, but this is universally applicable. 
7. It may be of advantage to reduce the index of the vari- 
able under the radical, which may sometimes be effected. In 
dec 4- 1 1 ”, assume x m+n + 1 = v n ; 
Then we have 
f'F ::r i: • * — j 
. dv . v n . (y K — i) m + ” 
And in the same manner surds of one order may be trans- 
formed into another. 
8. If the fluent be wholly logarithmic, we may assume for 
.jv,; Ol J. 
dx • ^x mJrn -f* 1 j >w = 
m + n 
irrationals of the first order 
FS %. ' 
' 4 ° 
air 
& 
noi 
. f R (a?) R (a?) . (a? -|- dfl . [oc fr) • • • 11 (a?) • (a? -j- * . -j- ^ 
i 0 
jF a ■: ;> f: ; , j f;~ : Ft : 
F. : : F FF .... F FF'.Fii 
v. : a 
13] s infij ©a 
m f ’ 
