* 
New Algebraic Series. 283: 
That is the quotientof the series fa, divided by /, is 
equal to the series raw _— a—benters, in the same man- 
neras a into fa,or 
m the second eivation ( Siomefas whine mb=a, 
we have b=—, whence ess “\ fa, or ms fa =f— 4043 
In the same manner we obtain (fa) =/(ay = 
fa, m. and n, being any two positive numbers. The equa- 
Basar ( fa)" =fma, holds therefore oe the positive 
Af m, be.a whole ey or a frac 
a like reasoni may. be saat ipioded that the 
same will hold when Sok value of m, is inco mmen- 
on We have algo] for'any porns. value of m,(fa)-" 
sia so aye 3 or, from: the. preceding, and, theo- 
rem (I1,) ( py Hi scp ooh ae e- Whence it 
follows that, whether m be agge a whole number ora 
fracti ion, positive or negativ , commen nsurable or incom- 
mensurable, ( fay" =fma, w will eis dyi hold, that is 
zz? 
(lta 4a(aPk) 5 re +&c. yr m+ ma (mak) 
+&c., whatever be the values of a and f, 
vot last equation, taking a=1, andk=—1, will be- 
2 i @ ao 
Co eee. 
ANA et rena = 
2° +&e. 
The formula for the Binomial therefore easily follows 
whatever be the exponent m- 
Taking k=o, ie 1,m=Az, the same eer will 
€come 
O4c$s45 ae 5 seeey "2144 see 
