923 



this form having the same meaning as the foregoing. But the latter 

 formula suggests the idea that we shall finally have 



«///,m = (a, 4 a, + a,)'«= [|(aj)^_^„, + a,\:,^a, + a,]'" , . . (82) 

 the last member of which points out the signification in a more 

 detailed manner. This is: expand the trinomial {'i ^ -\- (i^^ -{- (f ^)"\ as 

 if «7i,rt, ,<7.j were numbers; replace in each individual term of that 

 expansion, viz. 



Ca,9a,f'a,i , . ........ (83) 



where C is a whole number only depending on the exponents 7, A, z, 

 the exponent g by an index and in a^,cf{.i') the letter .v by a;-\-a^; 

 then expand every expression 



aj'ax^ffi^ -4- a,) 

 in a seiies of powers of a, and replace the exponents of a^ by 

 indices; in the functional expression represented by the now obtained 

 aggregate, or in the aggregate itself,^) replace the letter a' by .ï -|- (/,; 

 expand the product of the latter expression by a^' in a powder-series 

 of ^j and replace the exponents of a, by indices; then there results 

 an aggregate that, together with those obtained from the other terms 

 arising, like (83), from the trinomial (82), represents the required 

 function ani,m in the domain (oTj). 



We can easily see that the transition from (81) to (82) is allowed. 

 For looking into the matter thoroughly we see that the interpretation of 

 (82' may be obtained from that of (81) by changing the latter only so 

 far as to consider, in the development of the trinomial in question, 

 all terms involving the same power of a^ as an irresoluble whole. 

 It is not at all a matter of course that the two points of view 

 agree, nor even that the aggi-egates corresponding separately to 

 each of the terms mentioned converge. But we may again state 

 that the whole reasoning remains valid if we substitute for the 

 functions a^, a^, a^ their natural niajorants a/, a^, a^, and thus we 

 infer that there is no difference in the results afforded by the two 

 points of view. 



') See the foregoing footnote. 



