Eoss — Verh-Fiinctions. 



59 



or by dividing /? by 



f'+p,fi~+p,li~ . . ., 



since [/3'l[<A,.r = [[</>,.] [^f]]-'- 



We find 



([*.J-'r = ^" + — l-^)/" + ,772 



r + 1 V 



n j 



— ( 





r + S\ 



n j 



" {- 



_r-n 



r-\\ 



n 1 



r 1 





r + 2 



IB" 4 



n 



r-3 



" + . . . 



By putting -n, - r for n, r in either of these, we obtain the other. 

 Thus, if the original coefficients are the same, 



, . -1-1 

 that IS, <i) • i// = 1, 



which is verified by the evident relations 



»A„ = [<^-„l and = J /5-^ 



The series for [<^n]"^ and [j/'m]"^ can be obtained by other routes 

 than operative division — by differential means, and by obtaining a 

 general value for [</>„]'' and [i/^,,]*". So far as the writer can ascertain, 

 however, they are not generally known ; but the series for [</)i]"\ 

 giving a single root, has been previously obtained by the method 

 attributed to Lagrange and developed by Mui'phy, and also by way 

 of Lagrange's and Burmann's differential expansions. It is not the 

 complete invert of the original operation. 



So far as can be seen at present, no other series besides those given 

 above will fulfil the necessary condition that 



[<^.][(^„r and [c/>„]-^ [c^.] 



