16 'The Rev. B. Bronwin on the Inverse Calculus 



Change x into a—x, and we have 



{a — xY-<p {a-x) = f (E) {a - xf. 

 Therefore 



/* V-i dx (a ^xYfia-x) = <p (E) /* V-^ dx{a - xY 



*y <^ Q 



= f{E)ar-^-/^\--.dx{l -^)'' = r(«)f (E)a''+«j,p^^^ =Ha) 



suppose. Change a into «— ^, and we have 



T{n)f(,E)ia-xY'--^^^:^^ =Ha^x), 

 and 



rO^)'P(E) _y'"^^\. rxP-hlx[a-xY^''=r''xP-'dx'\;{a-x)', 

 or 



and 



r(w)r(p)f(E)a'-=r(«)r(;))fl'-<p(a)=^^)^"^^-'^7^4'(«-^). 



Change ^'"(p (.r) into <^ {x)^ and transform the second member ; 

 then 



as before. 



Resuming the equation 



T{n)<^{E)[a-wY^^ ^^}f) =^(«-^)' 

 we have 



or 



Multiply by a}-P=ia'^, and operate with (-7- j ; there results 

 r(«)r(;3)f (E)a'-/^= Cj-Yf v-^dv{l -v)P-'^av). 



