REPORT ON CERTAIN BRANCHES OF ANALYSIS. 217 



given tables of its logarithmic values to twelve places of decimals, 

 with columns of three orders of differences for 1000 equal in- 

 tervals between 1 and 2 * ; and similar tables have been given 

 by Bessel and by others. We may therefore consider ourselves 

 to be in possession of its numerical values under all circum- 

 stances, though we should not be justified in concluding from 

 thence that their explicit general symbolical forms are either 

 discoverable or that they are of such a nature as to be ex- 

 pressible by the existing language and signs of algebra. 

 The eqviation 



r (r) = (r - 1) (?• - 2) .... {r -m)r (r - m), 



r(r) 



or r (r — m) = -, tt-t h^ t— ., 



^ ' (r — 1) \r — %) . . (r — m) 



where m is a whole number, will explain the mode of passirig^ 

 from the fundamental transcendents, when included between 

 r = and 1, or between ?• = 1 and 2, to all the other derived 

 transcendents of their respective classes f. The most simple of 

 such classes of transcendents, are those which correspond to 



a) 



'/"•, 



which alone require for their determination the aid of no higher 

 transcendents than circular arcs and logarithms. In all cases, 

 also, if we consider r (r) as expressing \}[iQ. arithmetical y2Xvi& oi 

 the corresponding transcendent, its general form would require 

 the introduction of the factor V, considered as the recipient 

 of the multiple signs of affection which are proper for each dif- 

 ferential coefficient, if we use that term in its most general 

 sense. 



In the note, p. 211, we have noticed the principal properties 

 of these fractional and general differential coefficients, partly 

 for the purpose of establishing upon general principles the 

 basis of a new and very interesting branch of analysis |, and 



* Fonctions Elliptiques, torn. ii. p. 490. 



+ Thus,r(l.) = ^.,r(±) = i_v..r(A) = 2^ V.. 



\ 2 / 1 \ 2 / 1.3 \2/ 1.3.5^ 



V T, &C. 



% The consideration of fractional and general indices of differentiation was 

 first suggested by Leibnitz, in many passages of his Commercium Epistolicum 

 with John Bernouilli, and elsewhere ; but the first definite notice of their 

 theory was given by Euler in the Petershirgh Commentaries for 1731 : they 

 have also been considered by Laplace and other writers, and particularly by 

 Fourier, in his great work. La Tkeorie de la Projiagation de laChaleur. The 

 last of these illustrious authors has considered the general differential coeffi- 



