than the number of crossings. The sum of the products of the coefficients into the corresponding exponents gives, in each of the two parts of the symbol, double the number of crossings. These symbols contain the topologic character of any particular knotting. Listing next points out that the reduced three-crossing knot may be obtained by three half-turns about one another of two originally parallel cords whose ends are afterwards joined into one ring, and that the character depends upon the direction of the torsion. He proceeds to give a symmetrical knot, with seven crossings, in two different reducible, and one reduced, forms. The reduction of the first gives the three crossing knot, that of the second the four forms of the essentially non-clear coil of five intersections. (Figs. 6-9 of my first paper.) Their common symbol he writes as and he points out that, in this case, the Amplex belongs to each in succession of these four kinds of meshes. gives five different reduced forms, each with seven crossings; while the symbol But he adds the following extremely important remark :--“ In certain cases one symbol is equivalent to another, so that the reduced forms of the one can be transformed into those of the other." He states that this is the case with the last written symbol and the following: Thus there are, in all, eleven reduced forms of these kinds, all equivalent to one another, and all having seven crossings. The following are his figures: the first and second belonging to the former of the two equivalent symbols, the third to the latter He concludes this part of his Essay by saying that "these examples (confined as they are to single, closed lines), and the remarks made upon them, serve to show that the fundamental conception of twisting of lines is capable of being applied to the most complex space relations." It may be added that these very elegant and important results are given as statements merely, without any hint of how they were arrived at, or how they may be extended. In fact brevity has been sedulously studied, for all that is given about knots forms a comparatively small part of the whole of Listing's extremely valuable, but too brief, Essay. The rest treats, rather more fully, the whole subjects of inversion and perversion, screws of various kinds, plaiting and twisting, (with their applications to vegetable spirals, &c.), the numbers of lines joining given sets of points, the extensions of the meaning of the word Area, &c., &c. The above abstract, which contains almost all of Listing's remarks on knots, shows that he has long ago anticipated a very great deal of what I have lately sent to the Society. For myself, however, I may say that I have had to learn only two things (about knots) from Listing, viz. (1), a special case (which will be examined immediately), in which two forms are equivalent, though not transformable into one another by the methods given in my first paper; and (2), to value more highly than I had hitherto done the method of classifying forms by the numbers of each kind of mesh, and the right-handed or left-handed character of each. My first paper, as sent to the Society, was essentially confined (as indeed its title indicates) to the results deducible from a special elementary theorem,-one of two which occurred to me long ago when designing Vortex Atoms of various forms, and which I gave to Section A at the late meeting of the British Association. The second of these theorems (as will be seen by reference to my British Association paper, Messenger of Mathematics, Jan. 1877), was virtually the same as Listing's division of the meshes of a reduced knot into right- and left-handed,-only I called them black and white, but. as I did not see how to connect this theorem directly with the measure of beknottedness, I did not formally introduce it into my papers read to the Society. It is, of course, virtually included in the statements regarding coins thrown into the corners of cells,-for, taking the case of the silver and copper coins, the pair of left-handed vertical angles are those in which, or in one of them, there is silver-the right-handed, copper. Nothing can be clearer than Listing's statements on several parts of the subject: it is greatly to be desired that he had made many more. Still, with a cordial recognition of the great value of all that is to be found in Listing's paper, I adhere to what I said in my last communication, to the effect that the full character of a knot cannot be learned except from its "scheme," or something equivalent to it. That the type-symbol (when such a representation is possible) is ultimately equivalent to the scheme may possibly be true, especially when we consider that it virtually contains two independent descriptions of a knot (i.e. in terms of its right-handed and its left-handed meshes separately); that for purposes of classification it is superior is, I think, obvious, but I think it equally obvious that for the purpose of drawing the knot it is inferior. And the scheme for a reducible knot is quite as simple as that for a reduced one, while it is not easy to see what would be called the symbol of a reducible knot. Nor can I represent by * * (Added Feb. 7.)-I have just found symbols for which this is not the case. The following single instance is sufficient, for the present, to show that the type-symbol is not always equivalent to the scheme. The symbol may represent either a continuous curve with 7 intersections, or a complex system consisting of a circle intersected at six points by a skewed figure of 8. I shall discuss the subject fully in a paper “On Links," which I have in preparation. Listing's notation the double trefoil knot which has appeared in each of my papers; for, although irreducible (at least so far as I am aware), it contains several meshes which have angles of essentially different characters. Listing's avowed object was to simplify notation as far as possible. My impression is that, in one respect at least, he has carried simplification a little too far; for it cost me some little time and trouble to draw, from his type-symbol, the one knot which he speaks of, but leaves undrawn, viz., as above, Here is one of its forms: transformation will give the four others. In fact the type-symbol, even in this specially simple and symmetrical case, where it is much condensed, contains just as many separate typographical characters as the scheme; and I think there can be no doubt whatever that it is almost incomparably more easy to draw the figure from the scheme than from the symbol. Given the scheme, the symbol can be formed from it in a moment; while the finding of the scheme from the symbol is very troublesome. But in such a matter experience is the only guide, and I have had almost no practice in trying to draw from the symbol. Listing's type-symbol leads directly, however, to an inquiry not even suggested by the scheme; (for the latter, as I have given it, is essentially confined to a single closed curve),-viz., the forms of more than one closed curve, intersecting one another or not, which jointly divide an unlimited plane into given numbers of meshes with given numbers of sides. The first idea of this was suggested to me when I endeavoured to draw the curve with six intersections, whose type symbol is This symbol obviously satisfies the three numerical conditions; but, on trying to draw the corresponding figure, I found that it always came out as a species of endless chain of three separate links. One of its forms, from which the others can be found by transformation, is three circles, every two of which intersect. Two figures of 8, linked together at each end, give the symbol And by shifting the twist from one to the other, as explained in the latter part of this paper, the symbol may be changed to 7.4 +2,3 + p2 I have not as yet studied the theory of type-symbols, as it differs so much from my own method; but it is obviously desirable to find the criterion by which to distinguish from one another the typesymbols necessarily denoting one closed curve, and those necessarily denoting two or more intersecting curves. It is probable that there are symbols which may represent either kind of figure. The inquiry will no doubt be found very simple, if only approached from the proper side. I now pass to the sole point of Listing's paper which (so far as knots are concerned) was thoroughly new to me, though not unexpected; and I shall lead up gradually to the special case which he gives, using for the purpose the properties of the amphicheiral knots mentioned in my last paper. To apply the amphicheiral property to the production of new forms, we may begin by studying under what conditions the internal arrangements of a knot can be altered while four points of its contour, and two of the parts of the cord or wire joining pairs of these, are fixed. [The reason for the number four is, that when two only |