272 Mr. Evans on Newton's Binomial Theorem. 



In this series we may observe that its first term, P , is the 

 coefficient of «Q in the second term ; the second term, P"«Q, 

 is the coefficient of ^-Q in the third term ; the third term, 

 pV— . Q 2 , is the coefficient of n ~ . Q in the fourth ; the 



fourth term, PV^- . ^Q 3 , is the coefficient of ^-Q in 

 the fifth term, and so on, in like manner, ad infinitum. Now, 

 putting 



A = P", the 1st term, 



B = P w wQ, the 2nd term, 



C = PV-^- Q 2 , the 3rd term, 



D= P n n.^- . 2~ Q 3 , the 4th term, 



E - p n n . — . — . 'Z± Q 4 , the 5th term, &c. 



2 3 4 



Then, by substitution, we have 

 (P + PQ) n =P ? ' + »AQ+^BQ + ^CQ + -^DQ + 



JlZ±EQ + &c. 



5 



And, if we make n= — , whether affirmative or negative, 

 integral or fractional, we shall have 



(P+PQ)^=P^ + -^AQ + ^BQ+^CQ+ 



4'i on 



which is exactly Sir Isaac Newton's Binomial Theorem. 



For facilitating the exemplification of this celebrated theorem, 

 it may not be improper to observe, that P is the first term of 

 the binomial to be expanded ; Q the second term divided by 



the first, namely, -=— = Q-^> tne index of the power or 



root. 2. 



A = P m = 1st term of the series, 



B = — AQ = 2nd term, 

 it 



C = ^BQ = 3rd term, 



in 



D = m-2» CQ = 4tht 



E = 1Z** DQ = 5th term, Sec. 



XLVI. Re- 



