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XLV. An improved Demonstration of Sir Isaac Newton's 

 Binomial Theorem, on Fluxional Principles, more especially 

 calculated for the young Student in Mathematics. By the 

 Rev. L. Evans, F.R.S. arid A.S. $c. 



f ET ( P + PQ) re represent any binomial to be involved or 

 -" extracted, n being any power, or root, of the given quan- 

 tity P+PQ. 



Now, (P + PQ) n = F n x ( 1 + Q) w . 



And here we may preliminarily observe, that 



(1+Q) 9 =1 + 2Q + Q 2 , 

 (1+Q) 3 =1+3Q + 3Q 2 + Q 3 , 

 (1+Q) 4 =l+4Q + 6Q 2 +4.Q 3 +Q 4 &c. 



Hence, generally, we are induced to assume the rath power of 

 ( 1 + Q) by a series, as thus, 



(l+Qf = l+AQ + BQ 2 + CQ 3 + DQ 4 + EQ 5 + &c. the co- 

 efficients A, B, C, D, E &c. in the respective terms of the 

 series, after the first, to be determined hereafter. Let, now, 

 both sides of this equation be put into fluxions, and we have 



n . (1 + Q) w 7 l Q = AQ + 2BQQ + 3CQ 2 Q + 4DQ 3 Q + 



5EQ 4 Q + &c. 



Dividing both sides by Q, 



w.(l+Qf" 1 =A + 2BQ + 3CQ 2 +4.DQ 3 + 5EQ 4 + &c. 



This equation being divided by the assumed one, 



"hSLtS^Zl— A+2B Q.+ 3C<y+ 4D Q 3 +5EQ«+&c. 

 (1 + Q) n ~~ l + AQ+BQ*+CQ3+DQ*+EQ5 + &c. 



„ n A+2BQ+3CQ,'+4DQ3 + 5EQ«+&c. 



Or, 



l + Q 1-t- AQ+ BQi+ CQ.3+ DQ'+Efts&c. 



Multiply by the alternate denominators, 



n + n AQ + «BQ 2 + rcCQ 3 + wDQ 4 + wEQ 5 + &c. = 



and by transposition, 



+ mA^ -nB~) +nC") + nD} +") 

 n-A-2B V Q-3C VQ 2 -4D VQ 3 -5E VQ 4 - V&c.=0. 

 - A) -2BJ -3CJ -4D,) -j 

 By the lemma in Section XIV. of Simpson's Algebra, we 

 have 1. n — A = 0; 



2. nk— 2B- A = 0; 



3. «B— 3C-2B = 0; 



4. «C-4D-3C = 0; 



5. «D-5E-4D = 0;&c. Now, 



