35° Mr. Bromhead on the fluents of' 
algebraic fluxion, we may legitimately try <p (a?) -fi log. <p (*), 
h 
as a form to which the fluent may possibly belong.* 
2. It presents three cases: ist. where the fluent is wholly 
algebraic, for which we assume some expression so general, 
that its fluxion will include the given fluxion, if it admit an 
algebraic fluent ; or we find the fluent implicitly by an equa- 
tion : gdly. where the fluent is mixed, when we attempt to 
separate the algebraic part : gdly. where the fluent is purely 
logarithmic, when we assume, as in the first case, some ex- 
pression with indeterminate constants, sufficiently general to 
include the given fluxion. 
3. In assuming for an algebraic fluxion, it must be observed, 
that the fluent will always be a surd of the same order as the 
fluxion. On this principle Waring gives some assumptions 
for surds of the second order, but nothing has been attempted 
generally for surds of all orders, for want of some definite form 
which should include them all. In irrationals of the first order, 
the fluxion may always be reduced to series of terms, such as 
ay . jx -f ay . . . 
where the factors are all different, and where the indices are 
positive, negative, fractions, integers, or unity. Then let 
. i-.- f '.y, c . ^ -'TT . r V 1 ^ ;; p Ctf) i i‘ f: $ :r i p* f-jj (f ■ ■ 
R (x) be any expression cx n ~ l + cx n -” 2 -f . . . with inde- 
n—l 913flW «£92fiO ; 9T£ 013 fit <6 sss | ffcH ■ T 
terminate coefficients. Assume for the fluent 
■3W 
1 
te + . [x + . . . (x + 
R (x) 
n—i 
T 
\ 
' 
* It 5$ obvious, that the fluent of an algebraic fluxion cannot be of the form 
U88B 08 
? (x) + <P (x) . log. p ( x ), for its fluxion dp ( x ) + dp (*) ., log. plx) -f px 
is a transcendent. 
dp {x) 
2 
p{x) 
