915] 
ON THE PARTITIONS OF A POLYGON. 
105 
r — 4 sides to be distributed between the eight intervals, and the number of ways 
in which this can be done is 
r — S.r — 2.r—l.r.r + l.r + 2.r + S 
5040 • 
But this number requires to be corrected for coalescences, as in the case Three 
diagonals, A ; and the required number is thus found to be 
r — S.r — 2. r — l.r.r + l.r+2.r + 3 Q r — S.r — 2 .r — l.r.r + 1 
5040 d TlO 
Q r — 3. r — 2. r — 1 r — S 
+ d g 1 ur- 
Four diagonals, B.—There must be outside of three of the diagonals, that is, in 
each of three of the eight intervals formed by the diagonals, two sides; there remain 
r — 6 sides to be distributed between the eight intervals, and the number of ways 
in which this can be done is 
r — 5. r — 4. r — S.r — 2 .r — l.r.r+1 
5040 • 
There is a correction for the coalescence of two of the diagonals, giving rise to a 
form such as Three diagonals, B; and consequently there is a term 
r — 5. r — 4. r — S.r — 2 .r — 1 
120 ’ 
which, with the first-mentioned term, gives the required number. 
Four diagonals, G.—There must be outside of each of the diagonals, that is, in 
each of four of the eight intervals formed by the diagonals, two sides; there remain 
r — 8 sides to be distributed between the eight intervals, and the number of ways in 
which this can be done is 
r — *7 .r — 6. r — 5. r — 4.r — 3. 
2 . r — 1 
5040 
which is the required number. 
20. In the expressions of No. 14, A, 2A, 3A + 2B, 4A + 8B + 2G, if we regard 
the terminals of the diagonals as given points, then (1) we have two summits, 
which can be joined in one way only, giving rise to the diagonal-type A; (2) we 
have four summits, which can be joined in two ways only, so as to give rise to the 
diagonal-type A ; (3) we have six summits, which can be joined in three ways so 
as to give rise to a diagonal-type A, and in two ways so as to give rise to a 
diagonal-type B; and (4) we have eight summits, which can be joined in four ways 
so as to give rise to a diagonal-type A, in eight ways so as to give rise to a 
diagonal-type B, and in two ways so as to give rise to a diagonal-type C; we have 
thus the linear forms in question. To obtain the number of partitions, we have in 
each case to multiply by r. To explain this, after the polygon is drawn, imagine 
C. XIII. 14 
r.