irrational functions. 
Its fluxion will be 
DR 
n 
L (x) . + aj . ^x + aj . . -f aj 
R (x)Ja 
1 ?- 
— R (x) . | (« + 1 ) . (x •+• • • (% + + (P + 0 • (•£ + j • • (x 4- <*j+ • j. 
— - 
multiplied by dx . (<2? -j- a)* . . . (x + &)* the original fluxion. 
5 n 
Now that the two expressions may be equal, the coefficient 
found above must be = 1 , or we must have 
DR (*).Q-R (a?) . Q' = SR (x)V 
n — 1 n— 1 t n—\ * 
where O and Q' are the expressions in the coefficient involv- 
ing a, a , a. By equating the terms in this equation, the inde- 
1 2 n 
il 
terminates c, c , c, See. may be found ; but the reduction will 
i 2 
often be impossible, as there are more equations to be satis- 
fied than there are indeterminates. 
4. If any index a, j 3 , 7 = — 1, the expression fails, and there 
is no algebraic fluent ; also Waring says, that if the dimen- 
sions of a fluxional coefficient be — . — 1, the fluent cannot 
be algebraic. n r lQ t «noiiOBTl 
k. If o (an be an irrational function, let % = lax . cp (x) 
; ynM (Vf v 
= Jdx . v ; then since R \ a?, v \ — °> there are cases, where we 
’ ' i ’ '■ 0 .'ui\zA *^tnap 3 teoa 
. SfJ .11' ■) 
can determine, R \ x } z [ =0. 
: -V- Sj X . ' T 
6. If (p (or), tt (a?) be irrational functions of x, we have 
/ dx . <p (x) — fdx . | <p (a?) 4* 7T (a?) j — jdx . 7 r (a?) 
lOOlTSlO^ckol iT.'-i 10 uv’- )P,LU. nfOlV’do 01 li '• 
Now let 7T (a?) be so assumed, that 
• See ^rkns. i^^.^Emerson’s Fluxions. 1 
MOI! V 
^ i | ® •-{ %\ 
)f X, 
