43 
of Edinburgh, Session 1872-73. 
Also ( b ) is obviously the coefficients of the powers of a in 
a (a +- 1 +- — 
V a 
for the several positive integral values of y. Call the term in a x 
in this, i.e., the coefficient of a x ~ 1 in + 1 +- , A^y, and 
that at x, y in the scheme (c) Q^y , then 
Q x,y Q x— 1, y — 1 — A x ,y — 1 • 
and thus 
P#, y — A. x , y "b Q#, y 
— A x , y + A#, y — 1 4- A# — 1, y — 2 ”b 
This points to a very simple way of constructing the values of 
P*, y from those of y . 
In scheme (b), add to the number in any position that im- 
mediately above it, and also those lying in the left handed upward 
diagonal drawn from the last named, their sum is the number in 
the corresponding position in (a). Thus 16 + 6 + 3 + 1= 2 6. 
If D refer to x and D' to y, we have 
p / 1 , 1 , 1 , V 
y — ( 1 + jy j)jy 2 " 1 " J) 2 D /3 ' ' ' ’ y » 
= (* + DD'-l) A " r ’ ! '\ 
It is to be observed that, since if one player wins the other must 
lose, P _ x , y is the number of ways in which a player may lose, 
when he is x “ up” and y u to play.” 
The number of ways in which the game may be drawn is also a 
solution of the same equation of differences ; but the limiting con- 
ditions are now obviously independent of the sign of x : and are, 
taking it positive, 
Ptf,y = 1 if X = y , 
Ptf, y = 0 if x > y . 
