214 THIRD REPORT — 1833. ' 



value of r ( "H" ) = V'"', by the aid of the very remarkable ex- 

 pression for TT, which Wallis derived from his theory of inter- 



^'(^)^° = Cr(0) . r(l).iZ!::=(_l)r_l.l.2..(r-l).^-r, 

 d X'' r (1 — r) 



whenever / is a positive whole number. 



Conversely also, 



rf-i . a;-i r (0) T, r (0) 



= — -^ . x", where — ■^—' = oo . 



dx-^ T{1) r(l) 



d-^ . x -i _ r(o) 



d a;-2 r (2) 



d-^ . a;-i _ r (0) 



d a;-3 r (3) 



d-T . a-' _ r(0) 



x\ 



d x-r V {r) 



the arbitrary complementary functions being omitted. 



(5.) The occurrence of infinite values of the coefHcient of differentiation will 

 generalljf be the indication of some essential change of form in the transition 

 from the primitive function to its corresponding differential coefficients. 



Thus, 



<f-i 1 r (0) „ , , ^ 



. - = — ^-1 . aO = log X + C ; 



dx-i X r(l) o -r . 



this last result or value of -~— :■ . a:" being obtained by the ordinary process 



of integration : and generally, 



^'_^^-^ = LM xr-i + c^'-'' + Ci ^'-' + &c 

 dx-r r (r) re*) r(7-— 1) ' 



the first term of which is infinite, in all cases in which r is not a negative 

 whole number, in which case it becomes equal to ( — 1)"'' 1.2... ( — r) x''-^, 

 the complementary arbitrary functions also disappearing. If we suppose, 



however, r to be a positive whole number, and if we replace frjvs • "^ ^Y 

 its transcendental value already determined, we shall get 



- , = Hog « + C !■ + „ + &c., 



dx-r r (r) '■ " ^ -I r (;• — 1) 



which may be replaced by 



^' =m {'»«'+ (-')'rW.ra-0 + c} +-£l^,',+ . . .. 



which is in a form which is true for all values of r whatsoever, and which 

 coincides, for integral values of r, with the form determined by the ordinary 

 process of integration. 



More generally, 

 d-r . x-» _ r (1 — r) ^~„+r ; C ■ A"-" , Ci ..r'-"- ' 

 .dx-r r(l—n + r)'^ r(r — n + l)T{r — n) '^ ' ' " 



which is finite, whilst r is less than n ; and when r and n ai'e whole numbers, 



p ffi f\ 



becomes = (— 1)'' . — i ^ '' a''-", omitting complementary functions. 

 1 ()i) 



