922 



completeness, is thus easily proved, the generalisation for 

 n components being at once evident. It remains only to combine 

 the two preceding formulae, so that the resulting coefficient allium 

 may be expressed in terms of the coefficients ^i ,„, r72,w, ^^,m- 



We consequently work out the right-hand member of (80) accord- 

 ing to the rule at the end of N". 31, and then tind the following 

 new symbolic formula which answers to the earlier form (70) of 

 formula (73j, 



m 

 aill,m — yi- ^njc g,'"-^- aji^k (■« + «,) .... (80') 

 Ü 



The right-hand member must be developed according to ascending 

 powers of a,; tfiat this power-series is entirely determined follows 

 from our detailed investigation in N*. 31, according to which the 

 function «//,,„ (.c) is regular in the domain («,) and therefore also in 

 («3). Furtlier the exponents of rt, have to be replaced by indices; 

 the so obtained aggregate converges, according to the investigation 

 mentioned, absolutely and unifornily in the domain («,) and the same 

 holds for each of the m -\- 1 aggregates that can separately be 

 derived from the members of the sum (80'): this latter assertion, 

 moreover, corresponds to an earlier stage of the interpretation of 

 (70). The function <///,,„ (.<) has been considered in this as a whole, 

 but now it must be determine*! by means of (79), making formula 

 (80') pass into 



in 



g/;/^„, = y^A-m^«,'"-^';(i«i|.r+^, f aj^lx+a, .... (81) 

 



with the same signification, provided the factor between braces 

 be interpreted as a whole, according to the rule prescribed for the 

 working out of (79), substituting at the end of the process x f a^ 

 for .t' ^). The latter formula we may write in a more simple manner thus 



«ƒƒ/ 



,„. = %!: w^a,"'-^(a, j-aj* (81') 



1) By the words at the end we mean: after the function an [x) having been 

 constructed as a whole. Formula (81) would not express the same thing as (80') 

 if we interpreted it such that in each element of the infinite aggregate that is, in 

 general, obtained from (71)) for on,k, ^ were to be replaced by ic -f a^. Mean- 

 while we may observe that the latter mode of calculating an^k might really be 

 applied, as it will be seen on noticing that the preceding investigations remain 

 intact if the coefficients «i, a.2, «3 be replaced by their natural majorants. This 

 remark may be of use in theoretical questions. 



