440 
/ F 
X 
v/ 
Mr. Vince's new Method of 
(by fubftituting for F its 
2 mx ia l x — a + b . nix’" 
n »/ « rn J 
2 v x ~ a xx b 
value in the latter quantity, and putting A, B, C, &c. for 
the co-efficients which arife therefrom) F x s/ x m — a x x m — b — 
a/ 
H . 
A" X 
y' m 
m / 
— a XX ~b 
/ n — m . _ n — zm . 
— ^ * .. -C A i_ ■ 
V 'x m -ax x m -b J 
a X x m —b 
&c.-/ w 
m y ™ 
zm — I . . / ?# — i . 
X 2 ”V — Now 
n . 
X X 
%^x a - aXx m -b 
m m j 
x — a X X — b 
n — m . 
X x 
x m — a x 
-i’ / 
fit . . ’ rrr ? x — Q. 
x — axx A * u 
x m -b' s/ 
n—zm . 
A? AT 
« — 2 W . 
AT 
x a ~oxjr-b x ~ a 
Apf, &C. 
y x m -b 
But 
A * — x n ~ m x + ax n ~ im x + a'x"~i m x 4- &c. + a' 1 x n rm x + 
a;”- 2 ’”* 4. ax”-i”x + &c. + tf’— 2 *’’ - ”** 4 
AT — 0 
a: — a 
n—zm . 
X X 
x — a 
& c. 
r n—r tn , 
ax x 
x — a 
r — i j^n—rm . 
^ x 
x —a 
®*-3"x + &C . + 4 
& C. &C. 
r — 2 n—rm . 
a x x 
m 
x —a 
Hence, by fubffitution, <- /\”,v — - = F x s/x*‘ - a x x m - b 
-A f, x n - m x\J x -~ - A« + B X r X n - 2m x\]^f-^ - Aa 2 + B a + C 
1 ^ — b x — b 
x Z' x*-i*x\ x ~~- ~ - &C. - A^- 1 -f Bn’- 2 , + C« r -3 + &c.~ x 1 
/ k f »» . /> » — m . 
x’ I '~ rm xy — A# r + Ba r_I + C a r ~~ 2 + &c. x/ — : 
-S w* 
im x x — a-kb . x gut 
2 V '>_«x*”’ 
Z 
