. ^.Jfrationa( functions. 33 7 
Cor. 4. Let the equation be 
(ax - a) + (bx - (3) . v » + (cx - y) .«*s 0 
from which we can determine 
6 -r — & _J_ /TE=r 
U 2 C^ — Zy — "V ( 2 cx — 27 
and therefore any fluent of the form 
J'dx . R |x, (• 
ax — a. \ n 1 
cx — yj 5 
j dx . R | X, n S/ /aX -|- (5 1 + V a.* X* -f- fix ■+■ y 1 
It is obvious that these deductions may be carried to any ex- 
tent, producing forms hitherto supposed impracticable. 
Cor. k. ^4 is both of the form K~\x) and R+ 1 (x). 
u a x -\- 0 
' 
Prop. II. 
W 
We can rationalize 
dx . R \x, R-'(i), R-R- , (x) ) ...R“ , R _, ...R _, R t: '(-i)} 
C , v J a J » n — 1 2. 1 
Let R -1 . . . R~‘(.r) = v, Then 
11 I 
, „ ' I = RR...R(f) . >tS 
R” v x == R . . . R (?) . 
>d alnetoifbos sifr* t 3ot£mmori9b- nommoD £ ot 
&c. = &c. 
which substituted in the original expression, make it rational. 
Cor. 1. If R = R = R, the fluxion becomes 
1 a n 
c ' K - ■■■'' T - tf| + «T- X . 
^r.R |x,R (i),R "(x), .. . . R (x)} 
1 * dli J9J .& .id’) 
Cor. 2. By this theorem, any of the expressions deducible 
from Prop. I. may enter contemporaneously, and we may find 
fluents of very great intricacy. 
Cor 3. The fluents 
I y - -- X X 
-*■ / a + b n V a. + 0x\— ■ 1 
