4?'] Vt'lsconsin- Academy of Sciences, Arts, and Letters. 



In tins expression we are to retain only those terms which have 

 positive indices. 



The binomial coefficient 



At = {y+t-D t =(''+^-l)^_i 



in this last form of the numerator arises from a finite number 

 {v—\) of factors, of which the iirst is y-\-t — l^ the last 

 t -\- 1. Furthermore, since it must be assumed that all 

 the exponents t, Ui, Uo, . • . finally become infinite, we 

 may write instead of 



the exi^ression 



(^-M-l)^_l 



(v-l)! 



where (r— 1)! is the product 1.2.3 (y—1). This change is 



merely the neglect of the lower powers of /' in comparison with 

 hiojher. At the same time the coefficients 



etc., are changed in value. Thas if s^7l denotes the sum of ^ 

 elements of the series w^, u^, u^ u^ the coefficient of the 



term z^ will be 



{y-i)\ 



y — 1 -^c /J „ \'y — 1 



t ^—^— 2 {t—s,uf~^ 4-:^( (5-5210 — .... c [^1 



— S2WJ ^— .... f 



in which ^ signifies the sum of all the similar values, and all 

 the sums 5 ,,2/ which are greater than ^ are omitted. 



In order tliat the coefficient A the term x^ may be found, 

 its value, expressed as a function of x, must be substituted for z, 

 and also the multiplication by the factors neglected above, 



must be performed, so that the power 2:' is changed into 



n _ p 



