J\ew Algebraic Series. 283 



That is the quotient of the series /a, divided by/6, is 

 equal to the series into which a — Centers, in the same man- 

 ner as a into fa, or b into fb. 



From the second equation (fb)'^=fmb ; taking mb=a, 



we have fc=— , whence (f—)=fa, or V fa =/— • (1\ ) 



In the sanie manner we obtain {fa)-r ='\/(/«)™ = 



/~^a, m, and n, being any two positive numbers. The equa- 

 tion (II), (fa)'^=fma, holds therefore whether the positive 

 value of m, be a whole number or a fraction. 



By a like reasoning it may be easily proved that the 

 same will hold when the positive value of w, is incommen- 

 surable. We have also, for any positive value of w, (/«)~™ 



1 > 



= -=j-j-r- : or, from the preceding and theo- 

 (/«) (» 



rem (IIJ {fa)-'''=Y=f{o-ma)—f{—m)a. Whence it 



follows that, whether m represents a whole number or a 

 fraction, positive or negative, commensurable or incom- 

 mensurable, (fa)'^=fma, will always hold, that is 



2 Z^ Z Z^ 



{\-\-a—-\-a{a-\-k) - — ~ -f&;c.)™ = lmaY+ma(ma4-^)'T — ^ 

 +<Stc., whatever be the values of a and k. 



This last equation, taking a = l, and A;= - 1, will be- 

 come 



m m m — 1 m m—\ m — 2 



^^+&c. 



The formula for the Binomial therefore easily follows 

 whatever be the exponent m. 



Taking k=o, a=l, z = ],m=A<r, the same equation will 

 become 



11 1 Ax Ax A^'x^ 



A^x^ 



1 .2. 3 "'' * * ' 



