258 EEV. T. P. KTP,K~ MAN - ON THE K-PAETITIOXS OF THE E-GON AND E-ACE. 
That is, 
R (8, 1)=2, E (8, 2)=4, E (8, 3) = 7, E (8, 4)=4, E(8, 6) = 4. E->(8. 1)=1, E’(8, 2) = 2 
R'(8, 3)=1, E’(8, 5)=1, E‘(8, 4) = 1, E'(8, 0) = 1. 
The singly reversibles E.(r, Tc) form seven classes, 
Ef(r, k\, k),, E“^(r, k\ Ef(r, k\ Er(r, k). 
LXII. We have now to investigate the number of irreversible (l-}-^)-partitions of the 
r-gon. And when we have determined ^)„, m>0, the equations of art. XXIII. 
will enable us to complete the tedious solution of our problem. It is necessary to 
demonstrate the theorems following. 
Theoeem V. Every \i-ly irreversible partition of an r-gon, if h>2, and every doubly 
irreversible partition in which a diameter of the r-gon is not drawn, is regularly built on a 
polygonal nucleus. 
For in each of the h sequences of configuration (art. III.) in the circuit of the r-gon, 
we see f marginal faces. These being all erased, we see an /-gon {f < r) with fh fewer 
diagonals, and still h sequences, for those of the r-gon have been treated alike. If this 
r'-gon is reversibly partitioned, it is not singly reversible, because it has repeated 
sequences (V.); therefore the theorem is proved by XII. ; for there is no drawn diameter 
by hypothesis. If the r'-gon be still irreversible, it has h! sequences {h!<^h), from which 
marginal faces can be removed as before ; and thus we shall finally reduce the figure, 
either to a reversibly-partitioned r"-gon, or to one having no marginal faces, i. e. to an 
;i-gonal nucleus. And by reversing the process of reduction, the r-gon can be regularly 
constructed on that nucleus. Q. E. D. 
LXIII. Theoeem W. If the nucleus line of a doubly irreversible r-gon built on it be 
erased, the figure becomes Ih-ly reversible or else Ih-ly irreversible, and has no di'awn dia- 
meter. 
For it has still two irreversible sequences, occupying each half the circuit of the 7*-gon, 
since the two sequences terminated by the nucleus-line have been treated alike by the 
erasure ; wherefore it can neither be oddly reversible nor oddly irreversible ( Obs. I, 3, IV. ; 
and 5, VII.). And evidently the r-gon cannot have a drawm diameter meeting its 
nucleus line. 
Cor. The r-gon, after the erasure, is one of those reducible to a polygonal nucleus. 
Theoeem X. If in any 2m-ly reversible, or 2va-ly irreversible, \-partition of an r-gon, a 
diameter be drawn which is not an axis of reversion, and which meets no diagonal, the 
figure becomes a doubly irreversible {l-\-\)partition of the r-gon, built on that drawn line. 
For such a dra^vn line disturbs the symmetry about every axis of a reversible, because 
it is (VI.) perpendicular to none of them ; therefore the result is not reversible. And 
every 2m-ly reversible or 2TO-ly irreversible has an irreversible sequence occupying half 
its circuit (III, V.), beginning at any angle not on an axis of reversion ; therefore the 
result is not singly irreversible. And as there is no other sequence terminated by or 
exhibiting a drawn diameter, since two cannot be dranm, the result has no sequence 
