of Definite Integrals. 17 



If therefore we change x^<^{fc) into <^{x\ we now have 



(3.) 







If we put D for ^, then 



dr 



and, as in (c), we have 



6''f^(^)=(p(D)£'-S s-'-'^^(^)=(p(-D)s-'-^ . . (r7.) 

 Or, if we put p* for the operation which converts e'"' intoa^e'*'', 

 and ^^ for that which changes s~'* into a^^s"'"'^, then 



and A: may be positive or negative, integer or fractional, or 

 any quantity whatever. I believe these forniulae are new, 

 and they admit of many uses. 

 Changing x into a + ^, we have 



and 



= r(«)f(S) ^^''""^Ha) suppose. 

 Changing a into a+x, this gives 



and 



' ^ ' 







or 



(-i)'r(w)r(;7)^(fl)s— = (-i)'r(w)r(p>— f(a) 



Change the function e-''*'(p(^) into <p(a;), and we have 



/ xP-^dx^{a + x) 



as heretofore. 



Phil, Mag. S. 3 . Vol. S 1 . No. 205. Jidy 1 84-7 . - C 



