PARTITIONS OF THE X-ACE AND THE X-GON. 69 
We may either consider P(#+1,f) to denote the tri- 
edral partitions of the z-ace which leave f pairs of con- 
tiguous edges of the z-ace undisturbed, or to denote the 
triangular partitions of the w-gon, which leave f angles of 
the 2-gon untouched by a diagonal. 
27. To show the use of these formule, we can begin 
at the beginning, thus: 
P(4)=1; 
a. P(5)—R(5,2)=1 : 17(56,2)=—0=RG6,2)—=R*6,2). 
b. B*(5+m,2)=0, if m>0, by (15)=0”. 
m—2 
ce. [(5+m,2=2 7, by 16 and a). 
d. Brite) = i (18.) 
d. R(+m, of QF _e (18 and a); m even. 
kaa=0. 
m—1 
d’, R(5+m,2)=2 ? , by (18 and a m odd. 
jax=1. 
25 m—2 
é. 1(5-+4+m,2)=2 DF rei atta: ee a (19 and a); in 
which, of course, m>1 and sre the second or 
third term must=O, as m is odd or even. 
As P'(5+4+m,2)=R*(5+m,2) evidently=0, we have, by 
addition of c, d, d’, d", e, 
A. P(54+m,2)= an +m,2) + R(5+m,2) +1(5 +m,2) 
m— m—2 ns 
=2 pol. 2 vag + 2n-1° oe 
where m>0O, and 2,,=1-—2,, , is a ceeniinee of the usual 
, mM. : ; 
form ;»?.¢. 3,1 or=0,, as | 18 or is not integer. 
This P(5+m,2) is the number of triangular parti- 
tions of the (5+m)-gon in which two angles, and two 
only, are untouched by diagonals, or the number of tri- 
edral partitions of the (5+m)-ace in which two pairs of 
rays, and two only, of the #-ace remain each a united 
pair. 
J. P(7,8)=P(4) =R*(7,3)=1, by (4,25). 
