39 
of Edinburgh, Session 1872-73. 
for we have obviously 
Po,i = 1 • 
Next, 
£d?x,x+2 = Px+2,ar+2 + P#+l,>+2 
= 2(x + 2) + (x -f 2) + (x + 1)(# -f 2) . 
?x,x+2 = |(> + 1)0 + 2) + |#0 + 1)0 + 2), 
no constant being added, for 
Po, 2 = 3 . 
Similarly, 
? x>x + z=^(x+l)(x+2)(x+S) +±(x+2)(x+S) + ±x(x + l)(x+2) («+ 3 ), 
for 
Po,3 = Pi , 2 + Po ,2 4- P— 1,2 = 4 + 3 + 1 = 8. 
F x, X + 4=|(* ■ + 2) (® ■ + 3) (x + 4) 4- + 1) (re + 2) (x + 3) (x + 4) + i x (x 4- 1) (x + 2 ) (x + 3) (x + 4) 
for 
Po,4 = Pi, 3 + Po,3 + P — 1,3 = 11 + 8 + 4 = 23. 
We may now, in conformity with these expressions, assume 
P*, x+ n = { K + — + 7 — — r r + . . . ]• + 1 . • .X + n 
( x x(x+l) ) 
Now, if y = x + n, the original equation of differences gives 
AP x ,x+n r= Pa?+2 , x+n + P#+l ,x+n 
where A refers to x and not to n. By the assumed value of 
Pff, x + n this becomes 
j~ (n + 1) A n 
L~ 
nB n 
(»- 1)0, 
J XX + 1 
x(x + 1) 
+ x(x + 1)04- 2) + * 
B n _2 
o_ 2 i 
| X 4- 2 
x + 2 
+ (x + 2) (a? + 3 ) + J 
Bn— l 
o-i | i 
X 1 
X + 1 
0 + 1)0 + 2) J 
VOL. VIII. 
