REPORT ON CERTAIN BRANCHES OF ANALYSIS. 209 



the principle of equivalent forms might be extended to this for- 

 mula (supposing it to be investigated for integral values of n 

 and r,) as far as the symbol n is concerned only. This arithme- 

 tical coefficient of differentiation (if such a term may be ap- 

 pHed to it,) will present itself in the expression of the rth dif- 

 ferential coefficient (r being a whole positive number,) of all al- 

 gebi-aical functions *; and it is for this reason that we aveojjpa- 

 rently debarred from considering fractional or general indices 

 of differentiation when applied to such functions, and that we 

 are consequently prevented from treating the differential and 

 integral calculus as the same branch of analysis whose general 

 laws of derivation are expressed by common formulae. 



But is it not possible to exhibit the coefficient of differentia- 

 tion under some equivalent form which may include general 

 values of the index of differentiation ? It is well known that 

 the definite integral 



/ d X \ log — ) 



(adopting Fourier's notation,) is equal to 1 . 2 . . . . 7z, when n is 

 a whole number ; and that consequently^ under the same cir- 

 cumstances, the coefficient of differentiation or 



/ d X \ log — ) 



n (rt — 1) . . . (w — r + 1) = 



^'^.(logiy ' 



and in as much as \hQ form of this equivalent expression is not 

 restricted to integral and positive values of r, we may assume 



• IT, -f 1 ^ f^' « (-l)'-.2.3...(r+l)a;' - 



* Inus if M = -; — ; — r, we have -. — 5 =: 7^ — \ — srrxi 



X (1 - ^ ('^ - ^) , r(r-l)(r-2) (r - 3) _ ^^ ] . 

 \ 2.3a;2+ 2.3.4.5 a;'' '/* 



I , d'u 1.2 r 



and if M = -TT. — I — 5;, we have 



V (1 + a;7 dxr — (1 — a"- ) »•+ 4 



X ( 1+ JL . '• (>• - ^) , 1 ^(,.-l)(r-2) (r-3) , & 1 

 12= x" "^ 2= .4= a-4 J 



+ This definite integral, the second of that class of transcendents to which 

 Legendre has given the name of Intecjrales Euleriennes, was first considered 

 by Euler in the fifth volume of the Commentarii PetropoUtani, in a memoir 

 on the interpolation of the terms of the series 



1 + 1x2 + 1x2x3 + 1x2X3X4 + &c., 



which is full of remarkable views upon the generalization of formula; and 

 their interpretation. The same memoir contains the first solution of a pro- 

 blem involving fractional indices of differentiation, 

 1833. P 



