finding Fluents by Continuation. 
441 
lm x x— a-\- b .mx‘ 
W X %/ X m — a X X m - b 
;-/w'x 
^ at — a X * 
x"-' rm x J llzf. ; hence, by fubftituting this for thelafl 
term, it is manifeft, that - W x v/ x«-ax ~FFTb will be de- 
ftroyed by the kft term of Fxv 7 x*—d x x" - b when we fub- 
m-j- I 
ftitute for F its value ; therefore, if we put M = ~ m+i +, 
we have J' x n x sj~~ — M x 
bx 
n — 
n— 2 ,m + 1 
+ &c. + 
V -r “ +I 
n — rm + 1 ’ 
s/x m -ax x"‘ - b -A J' x”- m x - Aa + B x J' x" 
* x —b 
Afifiya _ Jfif + Ba~+ C X I X n ~* m X J£lf. - 
Y m 1 ' J n — A 
Vtf 
X — o 
Aa r_I + + &c. - J x f x »- T ”xJlLj!. -Aa r + Ba ~‘ + CV ~ 2 
1 w*?; A 
+ &c. x / 
n — -r*« . 
# * 
l/‘ 
A’ — £ 
Hence if the laft two fluents be 
a; — a Xx 
given, we have the general law of continuation up to the 
fluent of x n x * . 
y x m -b 
The utility of finding fluents by continuation was manifeft 
to Sir Isaac Newton, who firft propofed it; and fince his 
time fome of the moft eminent mathematicians have employed 
much of their attention upon it. The method Which I have 
inveftigated and exemplified in this Paper I .offer as being en- 
tirely new ; and at the fame time it not only exhibits, at once, 
the general law up to the required fluent, but alio appears, from 
fome of the inftances here given, to be more extenfive and con- 
venient in its application than any method hitherto offered. 
The 
