222 THIRD REPORT — 1833. 



taining the general differential coefficients of other expressions, 

 such as (cos x)", cos tn x x cos n x, &c., which present no kind 

 of difficulty. In all such cases the complementary arbitrary 

 functions will be supplied precisely in the same manner as for 

 the corresponding differential coefficients of algebraical func- 

 tions. 



The transition from the consideration of integral to that of 

 fractional and general indices of differentiation is somewhat 

 starthng when first presented to our view, in consequence of 

 our losing sight altogether of the principles which have been 

 employed in the derivation of differential coefficients whose in- 

 dices are whole numbers : but a similar difficulty will attend 

 the transition, in every case, from arithmetical to general values 

 of symbols, through the medium of the principle of the perma- 

 nence of equivalent forms, though habit and in some cases im- 

 perfect views of its theory, may have made it familiar to the 

 mind. We can form distinct conceptions of m . m, m . m . m, 

 m .m .7)1 , . . . (r), where m is a whole number repeated twice, 

 thrice, or r times, when r is also a whole number ; and we 

 can readily pass from such expressions to their defined or as- 

 sumed equivalents m^, m^, ... tm'': in a similar manner we can rea- 

 dily pass from the factorials * 1.2, 1.2.3,... 1 . 2 ... r, to 

 their assumed equivalents r(3), r(4), . . . r(l + r), as long as r 

 is a whole number. The transition from m^ and r (1 + r) when 

 r is a whole nvimber, to 7n'' and r (1 + r) when r is a general 

 symbol, is made by the principle of the equivalent forms ; but by 

 no effort of mind can we connect the first conclusion in each case 

 with the last, without the aid of the intermediate formula, involv- 

 ing symbols which are general in form though specific in value ; 

 and in no instance can we interpret the ultimate form, for 

 values of the symbols which are not included in the first, by 

 the aid of the definitions or assumptions which are employed 

 in the establishment of the primary form. In all such cases 

 the interpretation of the ultimate form, when such an interpre- 

 tation is discoverable, must be governed and determined by a 

 reference to those general properties of it which are inde- 

 pendent of the specific values of the symbols. 



» Legendre has named the function r(l + r) = 1 . 2 . . . r, the function 

 gamma. Kramp, who has written largely upon its properties, gave it, in his 

 Analyse des Refractions Astronomiques, the name oifaculte nimierique; but in his 

 subsequent memoirs upon it in the earlier volumes of the Annales des Ma- 

 thematiques of Gergonne he has adopted the name oi' factorial function, which 

 Arbagost proposed, and which I think it expedient to retain, as recalling to 

 mind the continued product which suggests this creature of algebraical lan- 

 suaffe. 



