324 Mr. J. Herapath on the Binomial Theorem^ 



Now, if n = r-\- v, and 7ihe as before an integer, r or v may 

 be any number rational, irrational, or imaginary ; and since 

 the sum n of these numbers is an indeterminate positive inte- 

 ger, they will in point of value be independent. But 



71-1 n-2 , , ,r + i)— 1 r+v-2 



or 



n +an +("l ■■■ _ (r + t)) -|-a(r+f) +a,{r + v) 

 r.2. ..(<?-!) ~ 1.2. ..(y-1) 



= !7. + 2,(?-l),...?„ (F) 



And because in the two right hand members of this (F), r 

 and V are independent variables, these members when duly 

 reduced must not contain any product of the powers of the 

 variables ; for if they did, the function of either variable would 

 be affected by the changes of the other variable, which it 

 should not. The middle member therefore of (F) must con- 

 tain only the sums of the powers of r and v, and the right 

 hand member only the two exterior terms. Consequently 



^r^^v~ 1.2. ..(9-1) 



which again, on account of the independence of the functions, 

 will evidently give 



(^_l)(,-2)... _^ (,-i) (.,-2)... 



?r = '' 1.2...(,-1) ^"^ ?. - ^ 1.2..,_lT 



which completes the proof. 



This conclusion we may easily verify by actual reduction. 

 Thus 2 +2 =2, =2 = n = r + n 



r V r-\-v n 



which since 2 and 2^ are independent, as well as r and ?i, give 



2 = ?• and 2 = u 



and because 



(r + y) '"^""^ = 3 + 2- 2 4- 3 = 3 + i;?- + 3 



2 

 (>•- + f-) - (•'• + ") 



= 3+3 



O 1.2 



we have by reason of the independence of the functions 



3 = !-(f-J2 and 3 = ^-i^ 



>• 1.2 V 1.2 



and so it may be shown of the rest. 



We may here remark, that any coefficient of the binomial 



expansion, as y^, may be represented by A ~^r, so that we 



have the binomial theorem under the new form 



