336 Mr. Bromhead on the fluents of 
■ ' , 
It is thought unnecessary to prove, that the fluxions of all 
rational functions, and all rational functions of them, are 
themselves rational. 
Prop. I. 
dx .K]x, R V x ) } can be rationalized. 
, Let R""” 1 (x) = v ; a: = R (z; ) ; dx = dv . DR (v)* 
s * i 
which substituted produce the rational form 
dv . DR (V) . R { R (y ), r/} 
Cor. i. This form includes f1 : 
dx . R| .r, R"*" 1 (x), R R"” 1 (.r)j, . . . R ^x, R - " 1 (x) J j 
Cor. 2. We may find, a priori, what fluents will come under 
this form. For let x = R ( v ) any rational function whatever 
x 
Gt> C& . V C£/ . XJ **p ... 
101 
ST 
a + a . v -f a . v % + 
° * 
which is the general form taken by rational functions, when 
the integer powers are expanded, and the fractions reduced 
to a common denominator, the coefficients being positive, 
negative, or nothing. Hence we haye R - ” 1 (x) = v, deter- 
mined from this equation. 
(a . x — ■ ot\ + (a . x — aS . v + (a . x — a) . v* + . 
Cor. 3. Let the equation be 
/ -v * f 1 n\ * 
(ax — u) + (bx~—( 3 )v n = o 
Then R — 1 (x) = v — 
8 emouft .... 
1 
we know Jdx .R|x, ” } 
* See Note at the end of the Paper. 
