Here a, ẞ, y... are, in order, the several strings plaited-so that β, in the coil ẞ is the prolongation of a, y that of ß, &c., and a that of the last of the series. The expression α The expression means that a crosses over B. It is sometimes useful to indicate whether a crossing takes place to the right or left. This is done by putting + or – over the symbol. Thus the four crossings above may be more fully written as + − + − α The properties of this notation are examined in detail. It is shown, that the combination just written cannot be simplified in itself; but that are simply nugatory, and may be written off. But, on the other hand, the terms α β βα usually add to the beknottedness of the whole scheme. When the scheme is not compatible with a clear coil there occur terms of the form a a, and the application of this method becomes very troublesome. (a.) When the scheme has no merely nugatory intersections, the most complete knotting is secured by alternate crossings above and below; or, as we may write, AX BY C. . . . &c. and here there is no continuation of sign. (b.) Cases in which there is no knot at all may be obtained for any scheme by making each letter positive on its first appearance. The various coils are then, as it were, paid out over one another. This process will give rise, in general, to but few changes of sign: 2 K VOL. IX. but the number of such will usually depend upon the particular intersection with which we commence the scheme. Additional changes of sign, still without any knotting, may be introduced by various processes, of which the following is the simplest:-When two letters appear together twice, not necessarily in the same order, but with like signs, these signs may be changed. Thus, the following parts of a scheme and the statements already made about nugatory intersections can be applied to these and other combinations even when they occur separately once only in each of two separate knots on the same cord. This, and a great number of similar theorems, allow of a great special extension of the nugatory test already given-but an extension which cannot be made in any case until the signs of the intersections are given as well as the order of their occurrence. Again, though, as has been said above, continuations of sign இ disappear when an intersection is lost, it does not follow that if a scheme have continuations of sign it must necessarily be reducible. The annexed diagram is an excellent instance. Its scheme contains fourteen continuations, and only twelve changes, of sign, and yet the knot is irreducible. But if we suppose it cut across twice at the single unsymmetrically placed crossing, and the ends joined so as still to preserve continuity in the string, the scheme has still fourteen continuations, but only ten changes, of sign; and it does not involve any real beknottedness. The remaining figure illustrates a fully knotted scheme, where there are no continuations of sign, but in which the mere change of sign of one of the intersections produces four continuations of sign, and the whole beknottedness disappears. Similar remarks apply to most of the preceding figures. IV. A great many other deductions from the fundamental proposition are given-for instance, A closed plane curve, intersecting itself, divides the plane into separate areas whose number is greater by 2 than the number of intersections. Regarding the curve as a wall, dividing the plane into a number of fields, if we walk along the wall and drop a coin into each field as we reach it, each field will get as many coins as it has corners, but those fields only will have a coin in each corner whose sides are all described in the same direction round. The number of coins is four times the number of intersections-and two coins are in each corner bounded by sides by each of which you enter-none in these bounded by sides by each of which you leave. Cut off at any intersection and remove a portion of the curve forming a closed (not self-cutting) circuit. You thus abolish an odd number of intersections. Hence if there is an even number of coils, whether the whole be clear or not, there is an odd number of intersections, and vice versa. To form the symmetrical clear coil of two turns and of any (odd) number of intersections, make the wire into a helix, and bring one end through the axis in the same direction as the helix (not in the opposite direction, as in Ampère's Solenoids), then join the ends. [The solenoidal arrangement, regarded from any point of view, has only nugatory intersections.] An excellent mode of forming this coil is to twist a long strip of paper through an odd number of half-turns, and then paste its ends together, the two longer edges become parts of one continuous curve which is the clear coil in question. This result is applied to the study of the form of soapfilms obtained by Plateau's process on clear coils of wire. V. Another question treated is the numbers of possible arrangements of given numbers of intersections in which the cyclical order of the letters in the 2nd, 4th, 6th, &c. places of the scheme shall be the same as that in the 1st, 3rd, 5th, &c., ie, the alphabetical. Instances of such have already been given above. In the first of I (b), for example, the letters in the even places are DEABC. Here the cyclical order of the alphabet is maintained, but A is postponed by two places. Whatever be the number of intersections a postponement of no places leads to nugatory results. A postponement of one place is possible for three and for four intersections only. Postponement of two places is possible only for (four), five, and eight-three for seven and ten-four for nine and fourteen-five for (eight), eleven and sixteen,-six for (ten), thirteen, and twenty, &c. Generally there are in all cases n postponements for 2n + 1 intersections and for 3n+2, or 3n+1 intersections, according as n is even or odd. The numbers which are italicised and put in brackets above, arise from the fact that a postponement of r places, when there are n intersections, gives the same result as a postponement of n-r-1 places. [It will be observed that this cyclical order of the letters in the even places is possible for any number of intersections which is not 6 or a multiple of 6.] When there are n postponements with 2n+1 intersections the curve is the symmetrical double coil-i.e., the plait is a simple twist. The case with 3n+2 or 3n+ 1 intersections is a clear coil of three turns, corresponding to a regular plait of three strands. VI. Numerous examples are given of the application of various methods of reduction. For instance, the scheme AEBFC G D A E K F L G D H B K C L H | A which is rendered irreducible by changing the sign of B, is reduced by successive stages as follows: ABCGDAKLG DHK BCLHA, which is the simple irreducible knot of figures 1 and 3 above. 3. On the Distribution of Volcanic Debris over the Floor of the Ocean,-its Character, Source, and some of the Products of its Disintegration and Decomposition. By John Murray, Esq. Communicated by Sir C. Wyville Thomson. During the present session I propose to lay before the Society several papers on subjects connected with the deposits which were found at the bottom of the oceans and seas visited by H.M.S. Challenger in the years 1872, 1873, 1874, 1875, and 1876. Instruments in use for obtaining information of the Deposits. It will be convenient to introduce this first communication with a brief description of the instruments and methods employed on board H.M.S. Challenger with the view of obtaining information and specimens of these ocean deposits. The instrument in most frequent use was the tube or cylinder forming part of the sounding apparatus. During the first six months of the cruise this cylinder was one having less than an inch bore, and was so arranged with respect to the weights or sinkers that it projected about six inches beneath them. The lower end of the cylinder was fitted with a common butterfly valve. This arrangement gave us a very small sample of the bottom. In July 1873 this small cylinder was replaced by one having a two-inch bore, and it was also made to project fully eighteen inches below the weights. This was a great improvement, as it gave a much greater quantity of the bottom in most soundings. The tube was, in the clays, frequently forced nearly two feet into the bottom. On its return to the ship, the butterfly valves were removed, and a roll of the clay or mud, sometimes eighteen inches in length, could be forced from it. In this way we learned that the deeper layers were very frequently different from those occupying the surface. In the organic oozes-as the Globigerina, Pteropod, Radiolarian, and Diatom oozes-the tube did not usually penetrate the bottom over six or seven inches, these deposits offering more resistance than the clays and muds. Occasionally the tube came up without anything in it, but the outside was marked with streaks of the black |