-_—~ 
657). 
XXIII.—WNote on a Theorem in Geometry of Position. By Professor Tatr. 
(Plate XVI.) 
(Read July 19 ; revised November 13, 1880.) 
In connection with the problem of Map-colouring, I incidentally gave (Proc. 
R.S.E. 1880, p. 502) a theorem which may be stated as follows :— 
If 2n points be joined by 3un lines, so that three lines, and three only, meet 
at each point, these lines can be divided (usually in many different ways) into 
three groups of nu each, such that one of each group ends at each of the points. 
Fig. 1, Plate X VI., shows such an arrangement (drawn at random) with one 
mode of grouping the lines, indicated by the marks O, I, II. 
The difficulty of obtaining a simple proof of this theorem originates in the 
fact that it is not true without limitation. For it fails when an odd number 
of the points forms a group connected by a séngle line only with the rest, as in 
fig. 2; and, though we may enunciate the theorem in a form in which it is 
universally true so far as the literal interpretation of the words is concerned, 
we do not, so far as I can see, thereby facilitate the proof: while we deprive 
the theorem of its full generality. For the projection of a polyhedron cannot 
have a group of points joined to the rest by ¢wo lines only ; and yet the theorem 
is true for such a diagram. The altered form is as follows :— 
The edges of any polyhedron, which has trthedral summits only, can be 
divided into three groups, one from each group ending in each summit. 
But a diagram such as fig. 3, for which the proposition is obviously true, 
ig excluded from this enunciation, unless we agree to apply the term poly- 
hedron to solids such as (for instance) an ordinary cylindrical lens with two 
edges and flat ends. 
HamiLton’s Lcostan Game is a particular application of this theorem, the 
corresponding figure being a projection of a pentagonal dodecahedron. It was 
suggested to him by the remark, in Mr Kirxman’s paper on Polyhedra (Phil. 
Trans. 1858, p. 160), that a clear “circle of edges” of a unique type passed 
through all the summits of this polyhedron. 
In this note I sketch, each very briefly, a number of different ways of 
considering the question. 
1. The simplest mode is to join, two and two, in any way whatever, the 
points of the system, by lines additional to those already drawn, neglecting 
any new intersections which may thus arise. The figure has then an even 
VOL. XXIX. PART II. Ce 
