of Definite Integrals. [5 



the same result as in (1.). 



In (a.) change <p (x) into <p (-), and then x into—; we 



have 



• <^{x)=x-^<^{i-')\ {h.) 



and therefore 



(p(a4-^) = (a + ^)-"f(6-*'), 

 and 



r(?z)r(D-w) „ n / m •/ X 



Hence 

 and 



or 



r(«)r(;.) ^]5z^) ai-D^(,-o) =^ °° ^p-1 ,/^^(« + ^^). 



And, as before, 



( - 1)' r(w)r(p)a-»^(e-«) = ( - 1 )T(«)r(;;)(p(a) 



\daj J Q 



X 



a;''~*(/a;\|/(a + ^)j 



the same result as in (2.). 



We might by this method derive the forms of (p {a) given 

 by Mr. Boole; but my object is merely to show one use out 

 of many which may be made of the formulae (a.) and {b.) 



If Ar=l, and E=l+A; 'EJ'r=r-\-k,W'x'' =x'-x^. Giving 

 to k an infinity of different values, multiplying the results by 

 any constants, and taking the sum, we have 



x^(p{x) = (^{^)x'' (c.) 



It is plain that we may give to /^, not only integer values, 

 but fractional ones also, and any values whatever, and nega- 

 tive as well as positive ones; for the operation E''^ performed 

 on r, or on a*", merely changes them into r + k, and ^''^^ re- 

 spectively. The function (^{x) is therefore very general. 



