33 



reiluction of ( p') to (P) is made by writing for the quantity 

 jis in (P') that quantity divided by the root s to be ex- 

 tracted, or writing 71 for ns ; the reduction is therefore made 

 simply by the root s to be extracted, dividing ns by s, and 



writing the quotient for m; hence we extract the s root 

 of (P) by the same rule, that is, by writing for n in (PJ, 

 n divided by s ; hence 



n 



It matters not liow the s" root of the series of the form 

 1 + ax -\- bx^+ &c. can be extracted, or whether we should 

 have been able to accomphsh it if we had not known that 

 the series (P*) and {p) are represented by {i+cc)"' and (i+.r)". 

 By whatever process the / root of (P') is extracted, whether 

 discoverable or not, by the same process the s" root of (P) 

 will be extracted. The Binomial Theorem shews the series 



(P) to be the s" root of (P'), which is all we want to ascer- 

 tain. 



VOL. XII. 



