916 



together, and thus we may interpret the right-hand mentiber of (70) 

 as follows: replace it bv its expansion in a power-series of the letter 

 ft and substitute indices for exponents. That a uniquely determined 

 power-series coriesponds to (he right-hand member of (70) hardly 

 needs any further mentioning, this having alieady been stated for each 

 of the k -\~ 1 terms sei)arately. The marmer of grouping considered 

 here makes it clear, however, that the expression in question may be 

 transformed analytically before proceeding to its interpretation, owing 

 to the fact that a function in the neighbourhood of a regular point 

 can but be expanded in 07ie power-series. This remark will be of 

 use when A's of different indices are in some relation to each other 

 so that further reductions of (70) are possible. But a general reduc- 

 tion of (70) is not possible since in none of the k -\- \ terms of the 

 series occur terms with the same index at ).. 



But further symbolization of the formula for tk is possible if we 

 replace the index at the letter ). by an exponent; if, at the same 

 time, we omit for a moment the form (.?; -j- n) from ■?■, we way 

 write 



k 

 §i. = \7 Xv(^^M)^-'iS ...... (70) 







If this be interpreted such that, before performing other reductions, 

 the exponent of A be re(ilaced by an index and the form {x -{- ƒ*) 

 be added, the foregoing formula is produced again and there is nothing 

 to be established. But a new result is obtained if we do not consider 

 each of the k -\- \ nembers of the sum as a ivhoJe,h\\{ Qvery t^voÓlwqX 

 (.t--|-,u)^— '+i ;. , where X^ stands for )-i{x-{-n), as the sum of^ — z-j-1 

 magnitudes the symbolic representation of which is obtained by the 

 development of the binomial (.« -j- f t)^ -'+i and the multiplication of 

 each of its terms by A' as if A and [i were numbers '). The total 

 symbolic aggregate obtained in that way from (70') is an ending 

 power-series in X and ^j, so that any other development of (70') 

 than the special one mentioned leads to the same power-series. Now 

 the expression in question can be analytically reduced to {).-\-ii-\-x)^, 

 so that finally we have the symbolic form.ula 



§;i, = (A + /x + .^r)^ (71) 



The correctness of this interpretation of the product mentioned has been 

 pointed out at the end of the previous paragraph, and, as is evident from the expo- 

 sition there, the interpretation consists in considering the product as the symbolic 

 representation of the expansion in the Taylob series of T^\pc^—>Xi{x)\, A((x) being 

 the "origin" and x^^i the 'increment". This is contrary to what in N''. 24 led 

 to the formula of Bourlet, where we took Xi{x) as the 'increment" and u'i){x) 

 as the "origin". 



