658 PROFESSOR TAIT ON A 
number of points, with four lines drawn to each ; and can therefore be regarded 
as formed of superposed (not self-cutting) closed circuits, each of which cuts 
another in an even number of points. The new lines must be so grouped that 
in the circuits which contain them they alternate with lines originally in the 
figure. It will be seen in § 2 that this proves the theorem at once by the help 
of those circuits which contain none of the new lines. But the application of 
this method to particular cases is by no means easy; for we may have to try 
several combinations before we obtain a solution of the kind desired. 
2. Assuming, for a moment, the truth of the proposition as given in the 
first statement, it is obvious that the lines of any two of the groups together 
form a closed polygon or polygons, each of an even number of sides: and, 
conversely, when (as just shown) we have such circuits, the proposition is true. 
(The italicised words show at once the reason for the exception to the 
theorem. For if the single joining line be part of a polygon, that cannot be a 
closed one; and, if it be not part of a polygon, there must be at least two 
polygons with an odd number of sides each.) When there are more polygons 
than one, the letterings of the alternate sides of one of them may be inter- 
changed; and we thus get, by combining these separately with the third set of 
n lines, a couple of new solutions. If either of these consist of more polygons 
than one, this process may be again applied, and thus we have two more solu- 
tions. Hence it is always possible to obtain a solution in which two assigned 
sides of one compartment of the diagram shall form parts of the same even- 
sided polygon. (From this consideration, as appears in § 5, we have another 
direct proof of the theorem.) Hence, also, it would appear that, as this break- 
ing into different sets of polygons cannot go on indefinitely, there must always 
be at least one solution which consists of a single polygon: provided, at least, 
that we keep to projections of polyhedra, for the statement is obviously not 
true of diagrams like fig. 3. But on this point I am not yet certain; and I 
pass it by for the present, as it is not of importance to the proposition, though 
it would be of great consequence to the making a perfectly general puzzle on 
the plan of the icosian game. 
3. A glance at the groups of connected figures of Plate XVI. (in which 
the polygon or polygons are bounded by double lines), will show better than 
any words of description the nature of the processes which I have just indicated. 
Fig. 7 has a very large number of solutions, twelve only of which are drawn. 
,, 8 is merely fig. 1 a little distorted. The additional line, which distinguishes it from fig. 7, 
makes it essentially unsymmetrical. 
,, 9 is essentially the same diagram as that of the Icosian Game. 
,, 10 is merely fig. 3; with one additional line, causing one at least of the two-sided compartments 
to be joined to the rest by three lines This at once makes the solution with a single 
polygon possible. 
N.B.—When a figure is symmetrical about any axis, the perversion of any solution is also a solution. 
