913 



For the m^^^ derivative of 7\i we may as well give a symbolic 

 formula. We saw in the 4^'' communication that this quantity, 

 defined by Pincherlk by means of (45), may in the domain («) also 

 be found by formula (39) : 



7X"0(«) zzzz Vfc -^^^ ^ (39) 



^^ k! 



u 



of course for functions u belonging to (/?). Instead of this formula 

 we may write symbolically 



T('«)(i^)=:a'«u(^ + a) (68) 



which formula has (67) as a particular case {m = 0). This 

 might perhaps give occasion to make the mistake of substituting 

 in the factor a'" index for exponent before developing the form 

 u{x-\-a) in a power-series of the letter a; this should first be 

 done, then multiplication by «'" should be performed, and finally 

 exponents should be replaced by indices. 



We now come to the symbolic representation of the more general 

 functional theorem of Taylor, dealt with in the 4^'^ communication. 

 Applying (67) to the product of the functions v and u both belongijig 

 to (0) we get 



T(v{a)u{x)) = v{a;^a)u{.v^a) (69) 



provided no other meaning be as yet assigned to it than that the 

 right-hand member be regarded as a ivliole, according to which it 

 has to be replaced by the power-series in a which corresponds to 

 the function w {x ■■{- a)=iv {x -j- n) u {x -J- a), if a denotes a number. 

 This power-series, however, is to be obtained by multiplying the 

 partial series corresponding to v (x -\- a) and to u (x -j- a) according 

 to the well-known rule, and then ordering the resulting aggregate 

 so that terms involving the same power of n are combined. If, now, 

 we collect into one all terms of the aggregate containing the same 

 factor 



a"'M(»0(.f) 



mf 



which is due to the expansion of ii{x-{-a), the result for all values 



of m is the functional series of Taylor. For the whole of those 



terms corresponding to a definite value of m is represented by 



m! 

 which, through (68), is equal to 



is not omitted, as was the case with (23) and m.m. more general with all 

 symbolic expansions we shall treat of. 



