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TRANSACTIONS OF SECTION A. 399 
several writers. But Herr Waldemar Meissner, of Charlottenburg, has recently 
found that it is possible when p=1093, and for no other prime up to p > 2000. 
As it has been shown (by Herr Wieferich) that the possibility of Fermat’s Last 
Theorem «+ y’=2? required the possibility of 2”—2—0 (mod p’), one difficulty in the 
way of Fermat’s Last Theorem has now been removed. 
4. An Hlectromagnetic Theory of the Origin of Series Spectra. 
By Professor A. W. Conway. 
5. On Map-colouring. By Professor A. C. Dixon, F'.R.S. 
lf the surface of a globe (or a plane surface) is divided into provinces in any way, 
then by the use of four colours only, say, 1, 2, 3, 4, the provinces can be so coloured 
that no two which adjoin each other along a line are coloured alike. (This theorem is, 
I believe, as yet unproved.) 
Let a line in the figure be marked a when it separates two provinces coloured 2 and 3 
or 1 and 4, 6 when it separates 31 or 24, ¢ when it separates 12 or 34. Then the 
problem of colouring the map is equivalent to that of lettering the lines, so that at 
any “ vertex’ where three lines meet the three lines are differently lettered. (It may 
be assumed that not more than three lines meet at any vertex.) 
Let a vertex be marked + when a rotation in the positive direction about it leads 
from the a line to the 6 line and then to the c line, and—when this order is reversed. 
Let + 1 and — 1 be called the ‘affixes’ of the vertex in the two cases. Then the 
problem of colouring the map is further reduced to that of assigning the signs or 
affixes of the respective vertices, and the only conditions to be satisfied are that in 
each province the sum of the affixes of the vertices shall be a multiple of three : the 
provinces are supposed simply connected. 
These conditions are linear congruences to modulus 3. If there are p provinces 
and v vertices there are v affixes restricted by p relations of which only p—1 are inde 
pendent, since the addition of all the congruences gives an identity. 
Hence v—p-+1 of the affixes may be chosen arbitrarily and the rest found linearly 
in terms of them so as to satisfy the congruences. It is necessary, however, that no 
affix so found should have the value zero, and to prove that this can always be secured 
does not appear to be any easier than to prove the original theorem. 
The left sides of the congruences satisfy the following conditions :— 
(1) Each unknown occurs in exactly three of the congruences, its coefficient being 
-+-1 in each case. 
(2) The number of unknowns common to any two congruences is two or zero. (This 
excludes maps in which the same two provinces have two or more sides common. 
Such maps are easily reduced to simple ones.) 
Conversely, a set of linear expressions satisfying the conditions (1) (2) correspond 
to a certain definite map, drawn on a sphere or on a multiply connected surface or 
surfaces as the case may be. 
Since the theorem of the four colours is not true for maps on a multiply connected 
surface it must be impossible to deduce it from the mere general form of the con- 
gruences. 
The reduction from the four colours 1, 2, 3, 4, to the three kinds of line a, 6, c, and 
thence to the two affixes + 1, is curiously analogous to the process of solving a quartic 
equation by means of a cubic, which itself is solved by means of a quadratic. 
6. On a certain Division of the Plane. 
By Professor A. C. Drxon, F.R.S. 
7. A Development of the Theory of Errors with reference to Hconomy 
of Times. By M. D. Hersey. 
An enumeration of the results to which we are led in studying the problems 
of design and of computation is followed by a detailed consideration of the 
problems of observation. 
In connection with the problem of designing (or adjusting) apparatus sn as 
