EEV. T. P. KIEKMAJS" ON THE K-PAETITIONS OF THE E-OON AND E-ACE. 221 
polygon Q. The polygons Q,, Qg, the remaining portions of P, have disappeared in the 
erasure of pairs of faces about h and Q ; that is, every one of them, Q„j, is one of a set of 
two or more faces standing symmetrically about Q. Therefore Q is a nucleus, reversible 
about all the axes of the r-gon, free from diagonals, and having everything arranged 
symmetrically about it in the r-gon. Q. E. D. 
XIII. Theoeem K. If there he no axis of reversion in a partitioned x-gon except one 
undrawn, a, the figure is huilt symmetrically about a and about a polygonal nucleus P, 
which has either no diagonals, or only what are at right angles to a. 
This is established by the former part of the preceding demonstration. 
But the polygon P, in this singly reversible r-gon, if it has diagonals perpendicular to 
the axis, is not properly a nucleus ; nor is there any reason why one portion of it, Q, 
should be called a nucleus rather than Q, or Qj ; for P can be constructed by loading 
opposite sides of any of them with proper polygons. But if P is a simple polygon, it is 
properly the nucleus of the figure, which is made by loading opposite sides of it, on 
different sides of the axis, with the same polygons, thus preserving the reversibility about 
the axis. 
XIV. Theoeem 1j. If a diagonal axis of the nucleus is an axis of reversion of the 
x-gon, it is a diagonal axis thereof. 
For it passes through tAvo angles of the r-gon. 
Theoeem M. An agonal axis of the x-gon is an agonal axis of the nucleus. 
For a diagonal or monogonal axis of the nucleus passes through a vertex of the r-gon, 
and cannot be an agonal axis of it. 
Theoeem N. If an agonal axis of the nucleus is an axis of the x-gon, it may be an 
agonal, or a diagonal, or a monogonal axis of it. 
For the sides of the nucleus through which that agonal axis passes may be sides of 
the r-gon, or they may be loaded with agonally reversible polygons having the same 
agonal axis : this will then be an agonal axis of the r-gon. 
Or those sides may be loaded both with monogonally reversible polygons, having their 
axes in prolongation of that agonal axis : this makes it a diagonal axis of the r-gon. 
Or one of those sides may be loaded with a monogonally reversible polygon, and the 
other either not at all, or with an agonally reversible : this, if the axes of the imposed 
polygon are in direction Avith that agonal axis, makes it a monogonal axis of the 
r-gon. 
XV. Theoeem O. A monogonal axis of the nucleus may be either a monogonal or a 
diagonal axis of the x-gon. 
For the side of the nucleus Avhich that axis bisects may be unloaded, or loaded with 
an agonally reversible polygon, so as to make it a monogonal axis of the r-gon. Or that 
side can be loaded Avith a monogonally reversible polygon, so as to make the axis a 
diagonal one of the r-gon. 
It is easily seen, and unnecessary to be formally propounded, how and how far the 
converses of these latter propositions are to be laid doAvn. 
