[915 
ich X is 
obtained 
I- 
915] 
ON THE PARTITIONS OF A POLYGON. 
97 
This of course gives the series of identities 
2 
1.2 
X 2 = X 2 , 
073«' =0 X < X!) ’ 
6 
or say 
1.2.3.4 < x ‘>" = no x < X3 >" + io ( xs >! ; 
X 2 = X 2 , 
(X 2 )' = f X (X 2 )', 
(X 4 )" = f X (X 3 )" + {(X 2 )'} 2 , 
all of which may be easily verified. 
7. I multiply each side of the identity by x 2 , and write 
U 1 = x.X, 
1 
yi = j—2 ^ 2 > 
— ¿c 
Ui — x 
1.2.3 
1 
1.2.3.4 
(X 2 )", 
(X 4 )' 
V 3 = x 2 
V 4 = x 2 
Fi = a? 2 
1.2.3 
6 
1.2.3.4 
8 
1.2.3.4.5 
(X 2 )', 
(xr, 
(XT, 
We thus obtain two sets of functions U and V, satisfying the before-mentioned equation. 
We have 
(F a + yV 3 + y 2 V 4 + ...) = (Fj + yU 2 + y 2 U z + . ..) 2 ; 
and it will be observed that we have, moreover, the relations 
U 2 = ^x (x~ 2 F,)', U 3 = \x (x~ 2 V 3 )', U 4 = \x (x~ 2 V 4 )', .... 
8. In particular, if 
X = 
then the general term 
in X is x r ~ x , 
in X 2 is (r — 3) x r , 
1-x’ 
in X 2 is 
in X 4 is 
X 
r — 3 . r — 4 
1.2 
r — 3. i— 4.7— 5 
1.2.3 
the first term occurring when r = 3, 
» » ^ 4, 
» » r = 5, 
» » r = 6, 
C. XIII. 
13