which would correspond to

4+2,3

for the right-handed part, and would give us the form

or one of its deformations.

The criterion by which to distinguish at once whether such symbolic representations as those just given represent knots or links is easy to find. If we remember that each of the (even number of) crossings lying on a closed curve is a corner of one black and of one white mesh (contained within the curve)-while each of the crossings lying within it is a corner of each of two white and of two black meshes —we see that unless we can enclose a part of the graphic symbol in such a way that the sum of the exponents within the enclosure, and that formed by the doubling of the number of the joining lines which are wholly within the enclosure, and adding it to the number of those which cut the boundary, are equal even numbers—the figure is necessarily a knot. But if we can enclose such a part, it requires to be farther examined to test whether the figure consists of links or is a single knot.

Thus, in the example just given, the part

-13

is a simple oval divided by two intersecting chords into threecornered meshes-but in the following formula

seems to fulfil the conditions above, it does not represent a separ

ate closed curve. In fact, the upper line represents a crossing on the boundary, at which there is (internally) only a left-handed mesh, which is impossible if the boundary were a closed curve.

And the lowest line in the figure is a point in the boundary which forms a common vertex of three (internal) meshes, two right and one left-handed. This, also, is inconsistent with the boundary's being a closed curve.

There is only one other case which may cause a little trouble. It can easily be seen by the fig. of last page. For we may take out the following part of the symbol

12

which must obviously represent the lemniscate in the figure. Its exponents and lines do not satisfy our condition: but they will do so if we remove the diagonal line-which corresponds to what is (in the lemniscate when alone) a nugatory intersection.

I conclude by giving the representations, according to the method just explained, of some of the preceding figures. Thus the three first figs. of p. 325 are, respectively,

while the pair of common-symbol knots on the same page are

It may be observed that the present method gives great facilities for the study of cases in which knots are reduced, or are changed

into links, by the removal of an intersection. For, to take off an intersection is easily seen to be equivalent simply to rubbing out one connecting line in the figure, and simultaneously diminishing by unity each of the exponents at its ends. If it be the only line connecting these exponents, they are (after reduction by unit each), to be added together. And this consideration enables us to obtain, even more simply than before, the rules for distinguishing a knot from a link. I propose, when I have sufficient leisure, to re-investigate the whole subject from this point of view.

Meanwhile I may notice that it is exceedingly easy to draw the outline of any knot or link by this method. All that is necessary is to select a point in each of the lines in the figure, and join (two and two) all these points which are in the boundary of each closed area. The four lines which will thus be drawn to each of the chosen points must be treated as pairs of continuous lines intersecting at these points, and at these only. When there are only two sidesand, therefore, only two points—in an area, two separate lines must be drawn between them, and these must cross one another at each of the two points.

The annexed diagram shows the result of this process as applied to the following symbol

This method also clears up in a remarkable manner the whole

subject of change of scheme of a given knotting which was dis cussed in my last paper. To give only a very simple instance, notice that the first of the changes there mentioned is merely that from

where the double lines may stand for any numbers of connection whatever.

I conclude by stating, in illustration of the remarks made at the end of my last paper, that I have hastily (though I hope correctly) investigated the nature of all the valid combinations among 720 which are possible in the even places of a scheme corresponding to 6 intersections (only 80 of these are not obviously nugatory) --and that'I find only four really distinct forms. They are

1. Two separate trefoil knots. Here there are two degrees of beknottedness.

2. The amphicheiral form. (Figured on p. 295 of my Note on Beknottedness. Also in a clear form in the last cut of my first paper.) 3. Fig. p. 297 of the same paper. These two forms are essentially made up of a trefoil knot and a loop intersecting it.

4. The following knot, which belongs to a species found with every possible number of crossings from 3 upwards. This species furnishes the unique knots with 3 and with 4 crossings, and one of the only two kinds possible with 5.

there being n-2 lines in the lower group.

The three last forms have each essentially only one degree of beknottedness. In certain cases (see the foot note ante p. 296) we may give two degrees of beknottedness by altering some of the signs -but the knot has then one nugatory intersection, and falls into the class with five crossings.

A number of curious problems are suggested by the process which I employed in the investigation of these six-crossing forms. I give the following as an instance.

Take any arrangement whatever of the first n letters:- Say, for instance,

CNDA

LE.

change each to the next in (cyclical order, so that A becomes B, B becomes C, .. N becomes A) and bring the last of the row to the beginning. The result is

... ·

FDAEB... M.

After performing this operation n times we obviously get back the arrangement from which we started. [Thus in seeking all the different forms of knots of a given number of crossings, one alone of this set of n need be kept.] The problem is to find sets such that the original combination is repeated after m operations like that above. It is obvious that if m is to be less than n it must be an aliquot part of it, and thus n must be a composite number.

[April 11.-The references to Listing's type-symbol here given must be taken in connection with the extracts from his letter, ante, p. 316.]

3. Laboratory Notes. By Professor Tait.

(a.) Measurement of the Potential, required to produce Sparks of various lengths, in Air at different pressures, by a Holtz machine. By Messrs Macfarlane and Paton.

The general result of these strictly preliminary experiments appears to show that for sparks not exceeding a decimetre in length