XXIII.-Note on a Theorem in Geometry of Position. By Professor TAIT. (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 3n 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 n each, such that one of each group ends at each of the points. Fig. 1, Plate XVI., 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 single 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 two 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 trihedral 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, is excluded from this enunciation, unless we agree to apply the term polyhedron to solids such as (for instance) an ordinary cylindrical lens with two edges and flat ends. HAMILTON'S Icosian 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 KIRKMAN'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 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 interchanged; 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 solutions. 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 evensided polygon. (From this consideration, as appears in § 5, we have another direct proof of the theorem.) Hence, also, it would appear that, as this breaking 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. 4. Or thus: when a set of points are joined so that two, and only two, joining lines meet at each point, these lines must obviously form one or more closed polygons. Hence, in the case before us, by limiting the selection to two out of the three lines drawn to each point, we can always, in many different ways, form a polygon or polygons. If the number of sides in each of these is even, the main proposition is at once proved; for the alternate sides of the polygons belong to two of the three groups-the unused lines forming the third group. Such solutions must evidently be possible in all cases, with the exception of that already excluded. This knowledge, however, does not at once help us to a practical solution of the problem in any particular case. We must, therefore, look at the result more generally. If the selection we have made gives more than one polygon, two or other even number of them may have an odd number of sides each. Suppose there are but two. If these be connected by one line only, we have the excepted case above. If they be connected by three, or a larger odd number of lines, we may always proceed as is indicated in figs. 6. 6a shows the two odd-sided polygons. 66 and 6c show how, neglecting the points C and C', we form evensided polygons passing through them and including AB and A'B' respectively. Finally, 6d shows the result when the two latter figures are joined. Thus the proposition is proved by actually effecting the decomposition into polygons of an even number of sides. Hence it is true for any even number of points (the excepted case excluded) if it is true for smaller even numbers of points. But it is obviously true for two, for four, and for six, points. 5. Another mode of reaching the same conclusion, is to pass from a case of 2n points to one of 2n+2 by drawing a new line terminating in any two sides of one of the even-sided polygons of the former case (§ 2). That polygon remains even-sided, but its sides must be relettered; and then we have one or more solutions of the new case. In fact, by temporarily suppressing, two by two, points and their joining line (always taking care that the figure left shall not belong to the excepted case) we can reduce any case, however complex, to the four points for which the proposition is always true. [Or we may suppress one line, and divide the figure into two odd-sided polygons passing respectively through its ends. On restoring the line, these two polygons give a solution.] 6. Practically, in every case, the simplest mode of solution is to begin at any point, and go through all (through some, perhaps, more than once) till we return to the starting-point. Then treat, as not gone over, all the lines which have been gone over an even number of times. This process is very easily learned by trial, the only special rule to be attended to being that we must never isolate a point. Should two odd-sided polygons be thus obtained, we may either begin afresh :-or go over a second time, attending to the above rule, part of the region of the figure in which these two polygons are contained. It is easy to see the connection of this method with the idea of a galvanic circuit of unit strength circulating (say right-handedly) in each of the polygons and the treating of any new or unused line as a conductor which can, when necessary, be split into two traversed by equal and opposite currents. It is probable that the known laws of such currents in a network may lead to the proof of the existence of a single polygon when the figure is a projection of a polyhedron. 7. Another method is suggested by Mr KEMPE's solution of the map-colouring problem (Nature, vol. xxi. p. 399). As the number of districts is, necessarily, n+2, and the aggregate number of their sides 6n, there must always be at least one district with fewer than six sides. Now, one side may be erased from a district of two or of three sides, and restored again, without altering the nomenclature of the remaining lines. Similarly, either pair of opposite sides of a foursided district may be erased, and afterwards restored. But when we erase any two non-adjacent sides of a five-sided district, a condition is thereby imposed on the nomenclature of the remaining lines, with which I do not yet see how generally to deal. 8. An immediate consequence of the theorem is that, in any network of triangles (however many lines meet at a point) the sides of each triangle belong one to each of three groups into which the whole set of lines can be divided. The theorem itself follows, conversely, if this proposition be independently proved. 9. In No. 494 of the Astronomische Nachrichten, CLAUSEN has a problem closely connected with the present subject. It refers to the minimum number of separate strokes of a pen by which a given figure consisting of lines can be drawn. LISTING, in his Vorstudien zur Topologie, has shown how to find this minimum number by counting the points at which an odd number of lines meet. In our present proposition, if one polygon can be found containing all the points, it and one of the unused lines together form one penstroke, and the remaining group of 2-1 unused lines forms the rest. If there be two polygons, they and one of the unused lines together form one penstroke. And so on. 10. To apply the result above to the problem of map-colouring, insert a new district surrounding each point of the map where more than three boundaries meet. Then divide the boundaries, which now meet in threes, into three groups as above. (The excepted case obviously cannot arise). Now let O separate the colours A and B, or C and D ; I, A and C, or B and D ; and II, A and D, or B and C; and the thing is done. For we may now suppose the inserted districts to become smaller, till they vanish. XXIV.—On the Structure and Arrangement of the Soft Parts in Euplectella aspergillum. By Professor FRANZ EILHARD SCHULZE, Graz. Communicated by Sir WYVILLE THOMSON, V.P.R.S.E. (Plate XVII.) (Published by permission of the Lords Commissioners of the Treasury.) Although, from the careful descriptions which have been given by several competent naturalists, we may now consider ourselves tolerably well acquainted with the structure of the dainty siliceous skeletons of this and several allied Hexactinellid Sponges, this is by no means the case with their soft tissues; and the great cause of our imperfect knowledge of these interesting structures is that no observer has hitherto succeeded in procuring a really well-preserved Sponge of this group. It was with pleasure, therefore, that I accepted the offer of the Director of the Challenger Expedition to place some well-preserved examples belonging to different genera in my hands for investigation. Of all the specimens which have been sent to me, some fragments of Euplectella aspergillum in absolute alcohol are much the best preserved, and therefore the best suited for thorough examination. I commence with the description of the soft parts of this well-known and beautiful form, following the classification which has been adopted by Sir WYVILLE THOMSON in his descriptions of the species. 1. ON THE SOFT PARTS OF Euplectella aspergillum (R. OWEN). I received in January 1880, from the Challenger Office in Edinburgh, an entire specimen of Euplectella aspergillum preserved with its soft parts in methylated spirit, and six bottles containing fragments of the same species preserved according to different methods, viz.: 1. In picric acid. 2. In solution of acetate of potash, after previous treatment with osmic acid. 3. In chromic acid. 4. In glycerine, after previous treatment with nitrate of silver. 5. In absolute alcohol, after previous colouring with carmine. 6. In absolute alcohol simply. I will commence with a short abstract of previous communications on the subject by other naturalists. In 1868, after examination of dried fragments, Sir WYVILLE THOMSON characterised the soft parts of the Hexactinellidæ, which he supposed to be |