Mr. Evans on Newton's Binomial Theorem. 271 



Now, from these equations are derived the values of the re- 

 spective coefficients A, B, C, D, E, &c. thus, 



1. From n — A = 0, 



n = A. 



2. From nA— 2B— A = 0; 



nA-A = 2B, 

 (n— 1) A = 2B, 



^— • A = B, by substitution, 

 w.—ssJB. 



3. From mB-3C-2B = 0, 



(«— 2).B=3C, 



n— 2 



— - . B = C, by substitution, 



n— 1 n— 2 _ >-, 

 ~2~ * 3 "" ' 



4. From »C— 4-D — 3 C = 0, 



^— . C = D, by substitution, 



w— 1 n—1 n— 3 -pj 



2 * 3 ' ~T~ _ 



5. From tiD— 5E— 4D = 0, 



(»-4).D=5E, 



— — D = E, by substitution, 



n—1 n—1 n—3 n— 4 -r^ 



»._ ._._. — =E&c. 



Having, thus, found the values of the several coefficients, 

 A, B, C, D, E, &c. the assumed equation, by the substitution 

 of these respective values, will become 



(l + Q)" = l+f»Q + ».^Q» + »:^.^Q» + 



».^.^.^- 4 Q 4 +&c. 



Then P"x(l+Q) n , or 

 (P + PQ) n = P n +PnQ + P«.^Q*+P^^.^,Q' + 



„ n—1 n — 3 w — 4 /-w 4 . 0r „ 



Or, by writing the coefficients of Q more conveniently, 



(P + PQr==P n +P^Q+P M «Q^.Q+PV-f 1 .Q 2 .^Q + 



~n n—1 n— 2 ^^3 n— 3/-v . 



P «. -5- . — <?• — Q + &c. 



In 



