scraped to powder with a knife. They gradually get harder on exposure to the air.

The powder, heated in a closed tube, gives out water which reacts alkaline, and has an empyreumatic odour. Heated with strong hydrochloric acid, it liberates abundance of chlorine, and the residue which remains is white, consisting of silica, clay, and sand, the sand being the same as is found in the bottom mud from the same locality. Their composition varies greatly, different nodules containing different quantities of mechanically admixed mud, and the number of different elements found in them is very large. Copper, iron, cobalt, nickel, manganese, alumina, lime, magnesia, silica, and phosphoric acid have been detected in a large number; but I have not as yet been able to make a completed analysis of any of them. I have, however, made a few determinations of the most important component substances. For this purpose the outside and densest layers of the nodules were selected, and portions of them pulverised and dried for ten or twelve hours at 140° C. The amount of chlorine liberated on treatment with hydrochloric acid was determined by Bunsen's method, and the iron was determined by titration with stannous chloride. The samples analysed were from four different localities. Nos. 2, 4, and 5 were from the same place, No. 2 being the matter collected round a shark's tooth as nucleus; Nos. 4 and 5 being the outside rinds of ordinary nodules.

The results are given in the following table, the numbers being in many cases the means of several observations:

A is the residue which remains undissolved after treating the mineral with strong hydrochloric acid, evaporating to dryness and redissolving. In No. 5 it contains 85.16 per cent. silica, and in No. 6, 82.27 per cent.

B is the available oxygen determined by Bunsen's method.

C is the MnO2 equivalent to the available oxygen.
D is the MnO found by weighing as MnO4.

E is the Fe,O, found by titration with SnCl.

F is the alumina found by subtracting the Fe2O, found in E from the weight of the precipitate with acetate of soda.

G is the water expelled on ignition; it is obtained by deducting twothirds of the oxygen found in B from the loss of weight by ignition.

It will be seen from the results given in the above table that the nodules from different localities vary greatly in composition, though in the same locality they have similar composition, irrespectively of the nature of the nodules. The insoluble residue contains, besides silica and clay, sand of the same mineral nature as is found in the bottom at the same locality. The manganese is present wholly as MnO2, and the iron as Fe2O. In No. 6 there is 3 per cent. of cobalt; this metal, along with copper and a little nickel, is present in all of them. Zinc was not found in any of the above specimens.

2. Note on the Measure of Beknottedness. By Prof. Tait.

In drawing the various closed curves which have a given number of double points, I found it desirable to have some simple mode of ascertaining whether a particular form was a new one, or only a deformation of one of those I had already obtained. Of course the schemes (as described in my former paper) contain the desired information, but it may sometimes be difficult to obtain in this way; for, when the number of intersections is large, we may have to change the crossing which is taken as the initial one several times before we hit upon the same notation for like crossings (if such exist) in the two schemes compared. And the methods of deformation already given often present their results in forms so distorted that it is not easy at once to recognise their identity with other drawings of the very same curves.

The method of treatment described in my paper, which depends upon the study of the plait, supplies (by the + and signs over the various crossings) exactly the sort of information we require, though it may leave ambiguities. But some simple mode of applying it is requisite.

I first tried a modification of the process (formerly described) of

going round the curve and pitching a coin into each field or cell as it is reached. To make the required distinction between crossing over and crossing under, we may suppose the two coins to be of different kinds,-silver and copper for instance. Let the rule be :silver to the right when crossing over, to the left when crossing under. Then, however the path be arranged, of the four angles at each crossing, one will have no coins, the vertical or opposite corner will have two silver or two copper coins, the others one copper or one silver coin each.

It is easily seen that a reversal of the direction of going round leaves the single coins as they were, but shifts the pair of coins into the angle formerly vacant; also that in the deformed figures the circumstances are exactly the same as in the original. Hence we may divide the crossings into silver and copper ones, according as two silver or two copper coins come together. And the excess of the silver over the copper crossings, or vice versa, furnishes an exceedingly simple and readily applied test (not however, as will soon be seen, in itself absolutely conclusive of identity, though absolutely conclusive against it), which is of great value in arranging in family groups (those of each family having the same number of silver crossings), the various knots having a given number of intersections.

I soon saw that this process, so limited, was intimately connected with that required for the estimation of the work necessary to carry a magnetic pole along the curve, the curve being supposed to be traversed by an electric current, and it occurred to me that we might possibly obtain a definite measurement of beknottedness in terms of such a physical quantity: as it obviously must be always the same for the same knot, and must vanish when there is no beknottedness. The measure may be made more complete by recording the numbers of non-nugatory silver and copper crossings separately, with the number to be deducted as due merely to the coiling of the figure. I shall recur to this point later.

When unit current circulates in a circuit, the work required to carry unit pole once round any closed curve once linked with the circuit is 4. Instead of the current we may substitute a uniformly and normally magnetised surface bounded by the circuit. The potential energy of the pole in any position is always measured

by the spherical opening subtended by the circuit; but its sign depends upon whether the north or south polar side is turned to the pole. Hence there is no potential energy when the pole is situated in the plane of the circuit but external to it, and the value is

2 when the pole just reaches the plane of the circuit internally. Gauss gave from these results the value of a remarkable double integral extending over each of any two closed curves linked together in space. Clerk-Maxwell (Electricity, § 422) has shown that this integral may vanish even for a complex linking of the two circuits; and a similar difficulty is met with in the single circuits with which we are now dealing, so that a special set of rules must be made for determining the beknottedness in terms of the silver and copper junctions. But the difficulty just mentioned leads, as will be seen, to a very curious result.

To construct the magnetised surface which shall exert the same external action on a pole as a current in any given closed circuit does, we may either suppose a surface extending to infinity in one direction (say, for definiteness, upwards from the plane of the paper), and having the circuit for its edge; or we may form, as in the figure, a finite autotomic surface of one sheet, having the circuit for its edge. The only diffi

culty in estimating the work lies in the definite statement of how the pole is to move along the curve itself. For, if its path screw round the curve, ± 4 must be added to the work for each such complete turn. As an illustration,

suppose we bend an india-rubber band coloured black on one side, as in the figure, so that the black is

always the concave surface, we find on pulling it out straight that it has no twist. If both loops be made by overlaying, when pulled out it becomes twisted through two whole turns. This is an instance of the kinematic principle that spiral springs act by torsion.

Perhaps the most simple definite condition is that which I first employed, viz., to make the pole move along the curve, keeping always in the osculating plane and on the convex side. But we have then to arrange beforehand what is to be done at a point of inflexion. A practical rule, however, may easily be given from the con-

VOL. IX.

2Q

sideration of the magnetised surface above mentioned. Go round the curve, marking an arrow head after each crossing to show the direction in which you passed it. Then a junction like the following gives + 2 at each time of crossing; or, if we use the infinite surface spoken of above, it gives + 4 for the upper branch, and nothing for the lower (which, on this supposition, does not pass through the magnetic sheet). Change the crossing from over to under, and these quantities change sign. The junction figured above would, in our first illustration, be a silver one. But a still simpler process is to go round putting a dut to the right after each crossing over, and vice versa. Silver crossings have two dots in one angle; copper one in each of two vertical angles.

Now, in order that our rule may be such as to give no work where there is no beknottedness, we must make the required expression such as to vanish whenever all the intersections are nugatory. Those which are nugatory only in consequence of their signs are in pairs, silver and copper, and will take care of themselves, as we see by special examples like the following, in which the reversal of one of the directions simply reverses the

signs. Hence the only part to correct for is that depending on the number of whole turns, and the sketch of the india-rubber band above shows that the work at the vertex of each such partial closed circuit is simply not to be counted-i.e., that the ± 4, which would be reckoned for each crossing by our rule, is to be considered as made up for by the corresponding screwing of the pole round the curve.

To illustrate the application of this process, let us consider again the two distinct forms with five non-nugatory intersections

(the first being a modified form of the "pentacle," the second, fig. 6