434 Wisconsin Academy of Sciences, Arts, and Letters. 



Furthermore, since it is required to find tlie number of com- 

 binations which, taken from the terms of the series, have a sum 

 p, we form the series 



na, na, — 1 ncr. — 1 net, 



l-y 1- y 1 y ^-y 



na., no'o— 1 ncxo — 1 nar- 



— r ^ — r — - — r • — - 



LIS] 



ri'x^ na^—X na^—\ na^ 



zrV ::; — r — - — r 



X •\-x -|-...+a:°4-...+a; -\-x 



and multiply these together. The coefficient of the term x^ 

 will be the required number. 



To facilitate the multiplication, let us first divide the series 



respectively, and put 



21 



V r 



Thus, (having thrown out the factors 



there results the series 



1 -f 2: + ^2 4- . . . + 2^i~l 



1+2 + 2:3 _|_ _, J^z' 



"8 



1+2+ 22 + ... +2^y"-l 



in place of which can be written the expressions 



) ) .... -• 



1— z 1 — z 1 — z 



