224 EEV. T. P. KIEKMAN ON THE K-PAETITIONS OP THE E-GON AND E-ACE. 
XXI. Before we can proceed to investigate formulae for the determination of these 
numbers, it is necessary, — 
Problem a. To find the 'k-divisions of the r-gon or x-ace. 
By the (l-j-^)-divisions of an r-gon I mean the entme number of ways in which k 
diagonals can be di’a-wn in it, none crossing another, all ways being different which 
occupy different angles 1 2 . . r of the r-gon. Thus there are five 3-divisions, but only 
one 3-partition of the pentagon made by drawing a pair of diagonals. And there are 
five 3-divisions of a pentace made by breaking it into three triaces, but these are all the 
same 3-partition. 
If we call the number sought of (l+^j-di'visions of the r-gon D(r, k), we can express 
it in terms of D(r', k'), / < r, and k' < k. 
For consider any diagonal dra\vn from any angle /3 of the r-gon, dividing it into a 
(3+A)-gon and a (r— A— l)-gon. This line b Avill be drawn in the (l-}-^’)-diAisions along 
mth every (l-l-£)-division of the (3+A)-gon, combined Avith every (^— ^)-di^ision of the 
(r — h — l)-gon, s diagonals being dra^vn on one side, and k — s — 1 diagonals on the other 
side of b. 
That is, 2^D(3-|-A, £)x I)(r — 1, k — s— 1) 
taken from £=0 to z=.k — 1, is the number of (l-J-^)-di\isions in which that line b will 
be seen. 
If, now, we give to h every value from A=0 to h—r — 4, we shall have counted ever)' 
(l+^)-division in which any line b appears that can be drawn from that angle j3. K we 
put for (3 each of the r angles in succession, that is, if we multiply by r, we shall have 
enumerated every (l-l-^)-division in which any hne b appears, that is drawn fr’om any 
angle /3. But we have thus handled twice, once from either extremit)% every line in 
every set of k diagonals; that is, we have counted every (l+^)-dirision 2k times. 
Wherefore the corTect result is 
2k.'D{r, ^)=r.2A{r>(3+A, g).D(r-A-l, k-^-l)}, 
Avhere every value of h from A=0 io h=r — 4 is to be combined with every r-alue of 
2 from £ = 0 to i—k — 1. If, then, we know these dhisions for all valires of r and 
k up to D(y— 1, ^~1), we obtain D(r, k) by addition. And as D(r’-1-0)=1, and 
D(r, l)=^y’.(9"~3), D(r, 2) is given, and thence D(r, 3), and so on, up to D(r, k), for 
any given values of r and k. 
XXII. To find D(r, k) in terms of r, for the general value of r, and for a given value 
of k, we write 
D(r, ^)=Ar^*-l-Br^*“'ff- . . -j-Lr-f-M ; 
for we know that this number is not greater than 3))}*'“^xf/I’+l \ that of all 
possible sets of k diagonals that cross or not. Then, from the 2^4-1 equations, 
D( ^+3,^)=A( ^+3)=*HB( ^+3)^*-*+..+M, 
D( ^-f4,/)=A( A:+4)^''+B( ^+4)^''+*+ .. +M, 
D(3^+3, ^)=A(3^+3)“-f B(3y^+3)^‘+>+ .. +M, 
