902 



with the restriction that, in nsing tiie scheme, condensed expressions, as 



d.vP J \dzJ 



are again replaced b}' their developments according to powers of z. 

 The scheme therefore remains for the present triply infinite. For 

 the raajorant-scheme'(56) a similar one as the foregoing is obtained; 

 it is not necessary to write this down. If in this last we give to u-' the 

 value a'j -f- « and to c a certain real positive value C, all elements of 

 that aggregate are real and positive. But if these elements are grouped 

 such that the elements with the same power ^^ of c; are added v^^e 

 get, as it follows from what we mentioned above, a convergent 

 series. Any grouping of this special aggregate gives consequently 

 the same sum. Again, for other values of x in the domain (a), and 

 for values of c with a modulus not greater than $ the elements of 

 both aggregates are in absolute value at most equal to the corre- 

 sponding ones of the special aggregate just mentioned. Thus the 

 elements of the schemes for the values of x and z mentioned, 

 form an absolutely convergent aggregate, the sum of which is 

 independent of the way of grouping; the same holds of course for 

 sub-divisions of both schemes. We therefore can also say that for a 

 given, arbitrary value of x in the domain («), either scheme produces 

 for a??,y grouping of the elements, 07ie and the same entirely definite 

 collection of values, when z takes all values in a certain neighbour- 

 hood of 2 = 0. 



This aggregate of values is consequently also identical with that 

 which was obtained for the special grouping mentioned above (com- 

 bination of terms with equal powers of z), and of which we said 

 that it forms a regular function of z in the neigbourhood mentioned. 

 As such an aggregate of values can only be represented in one way 

 by a power-series in z, the schemes may be analytically operated 

 on, that is all reductions which are valid in analysis, if c is a 

 number, may be performed. If such reductions be applied to sub- 

 divisions of both schemes, they again give rise to completely 

 definite functions of z, as will be clear at once after all that has 

 been said, and it need never be feared that for a combination of 

 the aggregates other than the one considered first, a functional 

 expression in z will be arrived at leading to another and conse- 

 quently wrong operative-series. 



If now the scheme (58) is added according to columns, we find the ex- 

 pression indicated by Bourlet for the resulting operative function, viz. 



