917 



the interpretation of which is implied in what precedes. We only 

 wish to call the attention to the characteristic fact that the letters 

 A and (i must not at all be treated in the same manner: first comes 

 the change of exponents into indices of P., then the same change 

 with regard to (i. 



Finally the last step : the determination of the coefficients nr,,, from 

 the quantities §/,. by means of (24). If we put in this formula the 

 right-hand member of (71) we find 



a.„ = Vfc m;t(-^)"'-'(A + |u + .r)^ (72) 







and there is nothing to be proved, if we substitute in each of the 

 ??i -|- 1 members of this sum separately for {^ -\- ii -\- oj)^ its proper 

 meaning. In order to get this latter we must expand the trino- 

 mial in its power-series in A and (i : each of the terms then 

 has its own real value as is explained above, and the same 

 therefore holds for the product of such a term by the factor 

 7n]c( — a;)"^~^. We thus obtain for each of the m-\-l members of 

 (72) an aggregate consisting of a finite number of elements each 

 of which is characterized by a definite symbolic power of A and fi. 

 The total number of elements arising from the in -\- 1 members is 

 therefore also finite, so that it forms a new aggregate that may be 

 arranged arbitrarily. If this be done in such a way that terms 

 involving the same powers of X and }i are collected — these may 

 be added analytically, the meaning of a product Xp^'i depending 

 only on the exponents p and q and not on its source — then we 

 obtain a power-series in A and f<. But the 5a;?ié power-series evidently 

 corresponds to all expressions which can be derived analytically 

 from the right-hand member of (72). Since, now, this latter is equal 

 to (A-|-ft)'", we may finally write 



a,, =(;i+,t)-=(i;.j,+,.. + ,x)'« (73) 



where the last member shews more explicitly the signification 

 which is to be assigned to the formula. This is as follows: expand 

 the binomial (A-|-/i)"' analytically in ils power-series in X and ii; 

 substitute indices for the exponents of A and in Xi{,x) replace .v by 

 .V -\- (x; again develop the so obtained functional expression in a 

 series according to ascending powers of ^ and finally substitute in 

 these powers indices for exponents. 



This is the symbolic formula we had in view, expressing the 

 coefficients am of the resulting series P in terms of the coefficients 

 ;.,„ and n,n of the components P^ and P,. In deriving this formula 



