38 
Proceedings of the Royal Society 
Let this number be represented by P^ y . Then obviously 
Par+l.y+l = Par+2,y + Par+l,y + Pa?, y • 
If 
V x - y = a x b>J 
be a particular integral, we have 
ah = a 2 -+- a + 1 , 
so that 
P*, ! , = SCa*(a + l + -|) ! ' . 
Now the conditions are obviously 
P x, y = 1 , if oc>y\ 
and 
P — x ,y= 0, if x>y . 
Failing in several attempts to determine fully the special form of 
P x , y from these conditions, I bad recourse to a graphical method, 
which will he given below. But before I do so, I take another 
mode of integration, which leads easily to special numerical re- 
sults. 
Suppose y = x + n , 
then the equation becomes 
AP*, x+n — P#+2, x+n 4~ P#+l, x+n 
from which it appears that if we can find expressions for P X:X+m 
and P#+i, x+m we can deduce by summation that for P x -i iX+m . 
Let us first put n — 0 ; we have 
AP#, # = P#-|-2, x + Px+l,x =r 2 , 
since, obviously, each of these quantities is unity. Integrating, 
we have 
P X,X ~ 2 # , 
no constant being added, since it is clear that 
Po,o = 0 . 
Again, by the fundamental equation, putting n = 1, we have 
AP^,a:-(-i — P x+2,x+l + P# + l,x+l 
= 1+ 2(® + 1) 
Px, x+1 = os + ^(^+1) -f- G 
= O + l) 2 = x[x + 1) + (x -f 1) 
