Prof. Tait on Listing's Topologie. 41 



come to the exit point or (without reaching it) return to our 

 starting-point, to try a new combination. For it is obvious 

 that, if we follow our rule, we cannot possibly pass through 

 the same trivium twice before returning to our starting-point. 



(15) This leads to a very simple solution of the problem 

 of Map-colouring with four colours, originally proposed by 

 Guthrie, and since treated by Cayley, Kempe, and others. 



The boundaries of the counties in a map generally meet 

 in threes. But if four, or more, meet at certain points, let 

 a small county be inserted surrounding each such point; and 

 there will then be trivia of boundaries only. These various 

 boundaries may, by our last result, be divided (usually in 

 many different ways) into three categories, a, /3, 7 suppose, 

 such that each trivium is formed by the meeting of one 

 from each category. Now take four colours, A, B, C, D, and 

 apply them, according to rule, as follows; so that 



ol separates A and B or C and D, 

 /3 „ A and C „ B and D, 

 7 „ A and D „ B and C, 



and the thing is done. For the small counties, which were 

 introduced for the sake of the construction, may now be made 

 to contract without limit till the boundaries become as they 

 were at first. 



The connexion between these two theorems gives an excel- 

 lent illustration of the principle involved in the reduction of a 

 biquadratic equation to a cubic. 



Kempe has pointed out that four colours do not in general 

 suffice for a map drawn upon a multiply-connected surface, 

 such as that of a tore or anchor-ring. This you can easily 

 prove for yourselves by establishing one simple instance. (This 

 is an example of a case of Listing's Census.} 



(16) From the very nature of our science, the systems of 

 trivia, as we described them in § 14, may be regarded as mere 

 distorted plane projections of polyhedra which have trihedral 

 summits only. There are two obvious classes of exceptions, 

 which will be at once understood from the simple figures 7 

 and 8. Their characteristic is that parts of the figure con- 

 taining closed circuits (i. e. faces of the polyhedron) are con- 

 nected to the rest by one or by two lines {edges) only. The 

 lines are always 3n in number, and, excluding only the first 

 class of exceptions, can be marked in 3 groups a, /3, 7, one 

 of each group ending at each point {trihedral angle). 



Now in every one of the great variety of cases which I 

 have tried (where the figure was, like fig. 6, a projection of a 

 true polyhedron) I have found that a complete circuit of edges, 



