222 EEV. T. P. KIEKMAH ON THE K-PAETITIONS OF THE E-GON AND E-ACE, 
I)ef. A clear axis of reversion meets no diagonal at right angles. 
A scored or loaded axis of reversion meets one or more diagonals at right angles. 
An e-scored axis of reversion meets e diagonals at right angles, of course bisecting 
them. 
The clear and the loaded axis are both supposed undrawn. 
XVI. Theoeem V. If a clear axis of a {2m-\-2>)-ly reversible r-gon be scored in any 
way by e diagonals at right angles to it, the i-gon becomes singly reversible about that 
scored axis. 
For, let ABC.. be the axes of reversion of the r-gon N. The scored axis A remains 
an axis of reversion, because the symmetry about is not disturbed by the perpendicular 
scores ; but none of the other axes BC ... is perpendicular to A (Theorem B) ; wherefore 
each meets singly all the scores upon A, and is no axis of reversion (Theorem D) of the 
scored r-gon N'. Let then, M, any other diameter of N' not amongst ABC ... be an axis 
of reversion ; it meets all the diagonals to which it is not perpendicular in pairs whose 
angles it bisects (Theorem D); wherefore these are pairs of equidistants from the centre ; 
now the intersections of all these, except the newly-added scores upon A, are on the axes 
BC... [Cor. Theorem D); wherefore M, passing through the centre and one of these 
intersections, is one of the lines ABC. . ., contrary to hypothesis, which is absurd. There- 
fore A is the only axis of reversion of N'. Q. E. D. 
XVII. Theoeem Q. If any loaded axis (A) of a {2m-\-2>)-ly reversible r-gon (X) be 
cleared by erasure of the diagonals perpendicular to A, the cleared figure (X') is singly 
reversible about that axis A. 
For, let ABC... be the axes of reversion of N, on which are the diagonals at right 
angles to them abc . . . forming a system of lines symmetrically placed about the centre 
(Theorem D). No one of these B, after the erasure of a from A, is an axis of reversion 
of N', because the diagonal b at right angles to B is not one of a system of lines s^unme- 
trically placed about the centre (Theorem D). Let then, M, any other diameter of X' 
be an axis of reversion of N' ; this line meets all the diagonals of N' not perpendicular 
to it in pairs, whose angles it bisects, wherefore these pairs are equidistants from the 
centre of N'; but all the intersections of these pairs lie on the lines ABC... (Cor. 
Theorem D) ; wherefore M is one of these lines ABC ... Q. E. A. Therefore A is the 
only axis of reversion of N'. Q. E. D. 
XVIII. Theoeem 'R. If a clear axis (A) of reversion of a ‘Im-ly reversible partitioned 
x-gon (N) he scored by perpendiculars to it symmetrically about the centre, the figure is not 
made singly reversible about (A) ; but if it be so scored wnsymmetrically about the centre, 
the scored figure (N') is singly reversible about that axis (A). 
For there is an axis perpendicular to A (Theorem B) about which the sjTiimetry is 
not disturbed by symmetric scores, ^. e. pairs of parallels to it equidistant from the centre, 
or a diameter parallel with such pairs. 
But when the scores are not such pairs, or a diameter and such pahs, the axis 
perpendicular to A is no axis of reversion evidently ; nor is any other diameter of N', 
