614 



Going on in this way we find, by indicating the result of 7?i-fold 

 repeated differentiation by T*'") 



00 







According to (39) we can consequently write 



a'„=7'("0(t;) (44) 



and for formula (42) 



T(.u) = \1'^D: (42') 







Since the symbol indicating the operation T("'i has arisen by ?/i-fold 

 repeated differentiation from the symbol for 7', we may call the 

 operation T("' the m''' derivative of 7"; Pincheri.k introduces this name, 

 though in another way. We further observe that the transmutation 

 D answers in the functional calculus to the function y=^x in the 

 theory of functions, because the derivative of D is equal to 1, that 

 of Z)'" to m D'»—\ and the [m -\- 1)''' derivative of D'" equal to 

 zero. In consequence of' this we may call the transmutation indicated 

 by a rational integral function of the symbol JJ a rational Integral 

 operation. Infinite power series in D represent then what we may 

 call a transcendental operation. 



The formula (42') is now clearly recognizable as a development 

 in the series of Taylor, that is a development of Tw in the point v 

 according to powers of the simplest operation D, calculated for the 

 increase u; we have to take the term increase yy/^A z^ in the meaning 

 of geometrical increase, that is multiplication by u. We may also 

 speak then of the development of Tw in a geometrical viciniti/ of 

 functional point lo = v. For v = 1 the series passes again into the 

 particular one found before; it appears in that case that the coeffi- 

 cient a,n is equal to T('«) (1), that is the m*'' derivative of T applied 

 to the functional origin iv {x) = 1. For this particular value of 

 iv [x) the derivative in question may therefore also be found by 

 means of the formula (24); for ?(; = ?;, as is to be found in Pincherle's 

 paper, and as we are going to verify presently, the following 

 generalisation of (24) holds 



a\u ^ 7'''«)(y) = 7'(.'c'«i') — miA-T(.r'«-iu) + . . . + (— 1 )'"./;« r(t') (45) 



From the proof we have given of the general development it 

 appears that it holds under the same conditions as the particular, 

 provided that the "beginning point" v{.i') and -"uicrease" u{x) belong 



