Prof. J. R. Young on the development of a Binomial. 269 



exception; but from the others the case of «=1 is excluded: 

 it is comprehended in the well-known form 



V n(n—l) n(n—l)(n—2) , .1 ,„ . 



0=l-n+ -i^ '--i ^3 + &c ' * « ( 6 



supplied at once by the original development. 



As in each of the above forms the final coefficient is equal 

 to unit, it follows that, by transposing the final term of (4.), 

 we shall have the expression 



(—)~.,-. + gfi^>y.- «(— '>(—*> S-'+fa. (7.) 



ton — 1 terms; and similarly for the forms which precede (4.). 



Now all these formulas may be converted into others of 

 greater generality by aid of the following theorem, viz. 



If it be true that 



n(n — l)„ n(n — l)(n—2) „ , . 



0= - n + - v -— — 1 2m 1 -J± L 3« + &c. (8.) 



throughout a certain range of consecutive positive integer 

 values of ?», then it is equally true that, throughout the same 

 range, the following more general form will have place : 



(■r + 3/3)™+&c; J 



and also that if (5.) be true, then will 



(— l) w 1.2.3. ..n=x n —n(x + l) n + -&- — '-{x + 2) n 

 W ( W -l)(n-2) (a?+3)w + &c> 



(10.) 



2.3 



be equally true, x and /3 being any values whatever. 



This theorem may be established either by common algebra, 

 or by the first principles of the integral calculus; the latter 

 being the shorter method, I shall employ it here. 



Suppose the formula (9.) to be true for some preceding 

 value of m, as m=p, so that we may have 



o^-^+^+^^+^-^y- 8 ) 



(.t+3/3)p + &c.; 

 then, multiplying by (p + 1) dx and integrating, we have 



c=^--^+^-+'-ii^i>(, +2 ^.- ''(''- 1 ^- 2 > 



(a: + 3/3)*+ ■ + &<:. 



