326 Mr. J. Herapath on the Binomial Theorem^ 



tional, irrational, or imaginary. We have hence a general 

 expression for cf. .r" whether r and n be possible or imaginary 

 numbers. Before, however, I make any observations on the 

 properties of the numeral coefficient n^ I shall proceed to dis- 

 cover another expression much more elegant and commodious 

 than our (3). 



Thus A.a-'' = (.r+l)"-x" 



A''..t'^' = (x + 2)''-2(^ + ir4^'' 



A^.a:' = (^ + 3f-3(^ + 2)'''t-3(x+l)"-/ 



and generally when n is any positive integer 



A"./ = (x + rj)"-«Cr+n-l)" + «'^Gr + «-2)''. . . (4) 



continued to the term next preceding that whose coefficient 

 becomes =0. But these coefficients are the same as the bi- 

 nomial, and have therefore the same proof for non-integer 

 values of n. Hence (4) is true, whatever be the value of n, 

 and we know it is true for all values of .r and v\ it is there- 

 fore true universally. 



I shall show presently how to deduce from this finite ex- 

 pressions for negative integer values to n. 



If we put .r=Owe obtain the celebrated expression for 



a". O", namely 



a". o"= n- n («-ir+ n'-=^ {n-2)\ . . (5) 



true likewise for all values of n and v, whether possible or im- 

 possible. 



This expression I sent between two and three years since 

 to an able mathematician as true for imaginary values of 7i. 

 It was of course given without demonstration, and hence he 

 thought it merely a definition. The investigation, or rather 

 the demonstration, I have now given it, will however, I pre- 

 sume, entide it as much to the appellation of a theorem as the 

 binomial expansion. 



I have hitherto proceeded in an algebraico-demonstrative 

 manner, which will easily lay open the process of the follow- 

 ing investigations ; I shall therefore be more concise in what 

 I intend to add in this paper. 



An easy management of (4) will bi-ing it under the form 



or, abbreviated, f-{t-\^-t£^Y.x' (7) 



which 



