and General Differentiation and Integration. 323 



Of the law of the exponents and the coefficient of the first 

 term it is commonly thought superfluous to offer any proof, 

 whatever the exponent may be : I shall therefore, and for the 

 sake of brevity, attend only to the coefficients of the second 

 and following terms. Divide now in each power of (A) the 

 several coefficients by the next preceding, and there will re- 

 sult : 



3.d_j.9d 4,th_i.3.d 5th'_^,i.ih 6"'-;-5"' (m + ])'h4.jn"' 



1st power 1 



2d ... 2 



3rd ... 3 



4th ... 4 



5th ... 5 



6th ... 6 



nth ... n 



(B) 



n—(vi—l) 



If these quotients be nmltiplied together so that the 5'th co- 

 efficient shall be the product of 5-— 1 of them, we shall have 



(d: + 3/)"=x +nx y+n-^-x y^+n-^.— x f (C) 

 for whole positive numbers. 



Again, for non-integral exponents. 

 Suppose the second, third, fourth, &c. coefficients of the 

 rth power are represented by 2^ 3^, 4^, &c. respectively, in 

 which the figures have no meaning but as indices of the num- 

 ber of the coefficient. Neglecting then the xs and ys for bre- 

 vity, and multiplying the rth power by the uth, the actual ope- 

 ration gives 



1+2+3+4- +5... 



r »• r r 



1+2+3+4 +5... 



1 + 2„ + 3, + 4, + 5. 



2 +22 +2 3 +24 ... 



V V r ' V r ' V r 



3 +3-2 +33 ... 



V V r V r 



4 +42 ... 



I' ' V r 



+ 5 

 Hence 



2,..„=2^ + 2„, 3,^^ = 3^ + 2^2^ + 3„, 4, ^, = 4,. + 2; 3^ + 3^2^ + 4^ 



and generally 



!7,+. = !7r-|-2„(y-lV+3„(7-2),... .. (y~l)^2^ + r/,, (E) 



S s 2 Now, 



