142 PROCEEDINGS OF THE AMERICAN ACADEMY 



The fundamental equations are obviously 



F{i, j) = aF(i _ l,y) + (1 _ a) f{i,j), 

 f{ij) = bf{i + n,j _ 1) + (1 _ 6) F{<\j), 



in which ^ -\- n must be reduced to h whenever it exceeds h. Substi- 

 tution, transposition, and division give at once 



F{i,j) = AF{i - l,j) + (I - A)f{i + n,j - 1) 



= AF(:i — l,j) + BF(i-\-n,j—l)—aBFi^i-^n — l,j—l). 



When no discount is given, this equation becomes 



F{i,j) = AF{i -, 1,7) 4- BF{i,j - 1) - aBF{i -\,j- 1), 



which is an especial case of an equation solved by Laplace in his Cal- 

 culus of Probabilities. 



When it is a grand discount, or whenever 



i >- h — n, 

 the equation becomes 



F{iJ) = AF{i - l,i) + (1 - A)f{hJ - 1) 



= AF{i - \,j) + BF{h,j - 1) - aBF{h _ l,y - 1). 



These are special cases, whatever may be the discount : — 



F{1, 0) =/(/, 0) = 0, 



F(l, 1) = A, 



F{i, 1) = AF{i - 1, 1) ^ A'. 



In the case of the grand discount, 



nhj) -MJ -i) = A [F(i - i,y) -f{h,j - 1)], 



=Aii-f(h,j-in 



F(i,j) = A<Jr(l- ^)Ah,j - 1), 



F{i - l,i) = A*-' + (1 - A'-')f{Kj - 1). 



