344 
Mr, Bromhead on the fluents oj 
f 
dx . R 1 (ur), (p R 1 ( x ), $ R 1 ( x ), . . } 
for by taking R“ J (jr) = v, it is reduced to the former form. 
i 
Cor. i. If we can rationalize 
dx . R j.i?, R”* 1 R ( x ) | , we also can 
dx.n{x, R - ' (x), R — ‘ RR~'(i)} 
'* m a n m J 
Therefore we can find 
Jdx . R j.r, R~ (x), V a -f- h . R"'* 1 ( x ) -f ■ c . (R~~ l (a:)J 2 | 
Jdx , R ^ X 3 ^ a — J— fix "4" 'C u “h bx ^ &c. 
Cor, 2. Generally we can reduce 
dx . R | x, R R —I . . . R R“ ! ( x ) l to 
t o i M M- {-I ^ 
dx . Rj.r, R “ 1 . . . R(.r)j 
Cor . 3. In * <p (x), let v = (a;), and if it be an alge- 
braic function, R J, z>, j =0. Now take a: = R (z) 
C6 “f* CL Z "I” CL -j - &C» 
x 2 
a + a z + a z z + &c. 
Z 2- 
; and z> == R (2;) 
0 + @ z .+ &c. 
£ 
b -{■ b z + &c. 
with in- 
determinate coefficients. 
Hence we have R { R (z), R (2) } =0; remove fractions 
and make the coefficients of the powers of 2 vanish. This 
will give the indeterminates, if x and v admit common ration- 
alities. Thence we have dz . DR ( % ) . R (%) rational. 
z 2 
Should all the artifices in the foregoing propositions fail, 
we must attempt to resolve the fluxion into a series of terms, 
such that each term may be separately rationalized. This is 
