230 REY. T. P. KIRKMAN ON THE PARTITIONS 
triangle let all but y vanish, and in these y quadrilaterals 
let all the opposite edges vanish except z. If these opera- 
tions be repeated in the circuit of the r'-gon on the other 
side of the central quadrilateral, we shall complete 
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among which L’ will be twice constructed, i.e. with either 
face uppermost, if the spared edges, 
4(4-1)+z=4(7'- 6), or 
z=4(r'-k-5). 
If now, in each of these 
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figures, 
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r'-gons, 
we add 3a points in 3(7’—2) positions, viz. about the 
vertex of a marginal triangle, on a side of the central 
quadrilateral, and on the 4(r’—6) edges lying between 
those two positions; and if we repeat the distribution of 
4a points in order round the circuit on the remaining half 
of the r’-gon, we shall complete upon each 7'-gon 
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figures, 
which will all have a sequence repeated in the circuit. 
If the 7/-gon on which we operate has not two axes of 
reversion, that is, if z<4(4—1) or 7’ <2k+4, all our re- 
sults will be of the class I?(7,4+1), for the addition of 
the 2 points has added no axis of reversion to the figure ; 
but if the 7’-gon should have two axes of reversion, 7.e. if 
r=2k+4, the a2 points, being distributed in every way, 
may be so arranged as to preserve the symmetry about 
both axes, while they still exhibit a sequence repeated in 
the circuit. That is, we shall have constructed both twice 
every one L’ of the class °(7,4+1), which can be reduced 
by the vanishing of a points to the (24+4)-gon, and also 
once every one of the class R?(r,4#+1) which has a central 
face; that is, the product of the last written function of 
