ral value of the co-efficient of x can be no other than 



n. n — 1 n — r-\- X 



3 



ITi-EM ALITER. 



Let n and s be indefinitely great, whole, positive numbers, 

 80 that -7- may represent any fraction ; then by common 

 algebra 

 (i + ^)-= i+nsx + ns^-f^'+ScciP') 



{i + x)" = J+nx 4. n.^x^+ Stc. (P). 



Now it is proved in my Fluxions, that (i+a)— may be 

 represented by a series of the form 



1 + cf.r 4. ijr + &c. (P)— , where the form of the series in 

 respect to x is the same as that of the above series ; we have 

 therefore only to consider what is the relation of the corres- 

 ponding co-efficients. Now the series (P') and (P) are ex- 

 actly of the same form in every respect, the factor ns in the 

 former being represented by n in the latter. If therefore 

 we perform the same operation on these two series, the re- 

 sults must have the same form, and whatever change maj' 



take place on ns in (P'), the same must take place on ji in 



(P). If therefore we extract the s Root of (P') and (P), 

 the forms of the two series expressing the roots must be the 

 same, and the roots be deduced by the same rule. Now the 



