EEV. T. P. KIEKMAN ON THE K-PAETITIONS OE THE E-GON AND E-ACE. 259 
more than twice repeated, and is doubly irreversible. And evidently the figure is built 
on the drawn line, by applying to it the same polygon on both sides, so as not to give a 
reversible result. 
LXIV. Problem n. To find P(r, k)„, (h>l, n>2), the number of h-ly irreversible 
{l-\-^)-'partitions of the x-gon, which are built on the n-gonal nucleus. 
The construction difiers from that of reversibles, in that the -partitioned 
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(2+a„)-gon which is laid on the mth side of the ^^-gon in the first interval of ^ sides, is 
laid also on the m\h side of every interval, so that the series of loads imposed is h times 
repeated, and not reverted. The result will be an r-gon in whose circuit a sequence 
occupying ^ sides of the nucleus is h times repeated. 
The equations to be satisfied are 
rz=.n 
Jc—n 
+^ 2 + • • + 
(E.) 
exactly as in our previous constructions. 
We take any one solution, and lay on the polygons, and then make every possible 
change in the arrangement of the diagonals of the imposed polygons ; this gives us 
D(2-j-<Zi, ^j)D(2-j-<Z2? 02 ) . .^l^~\~an, 
\ h hi h 
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different sequences read in the r-gon upon the first ^ sides of its nucleus, which are all 
h times repeated in the same order in the circuit of the r-gon. We then take every 
other solution of (E.) that can arise from changing either the values or order of the 
numbers «2 • • 1 e^ e^..', and forming a similar product from each solution, products 
which will some of them be equal to each other, we shall have enumerated in the sum 
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2An, every possible sequence that can be read on ^ successive sides of the w-gonal nucleus 
h 
of any (Id-^’)-partitioned r-gon, which has that sequence h times repeated in its circuit. 
And no two of these sequences can be alike, because they are either made from different 
solutions of (E.), or have a different arrangement of diagonals in the ^ imposed polygons. 
Now every /w'-ly irreversible (l+^)-partition of the r-gon has and no more, of these 
sequences ; for it has a different ^-ple sequence of | 
^ successive vertices, and no more, reading always in one direction, because the sequence 
beginning at the vertex is identical with that commencing at the first ; and this 
?’-ple sequence of ^ vertices is h times repeated in the circuit, because a simple sequence 
2 M 2 
