A dew bow differs from a fog bow in being formed by minute globules of water resting on the ground; but these globules may or may not have been produced by the usual process of dew formation. In the Proceedings of the Society (vol. vii., 1870), Clerk Maxwell has published a note on a bow seen on ice. With the exception of this record, we have been unable to find any distinct account of observations of the phenomenon, although no doubt it must have been occasionally observed soon after sunrise.

The peculiarity of the bows we observed was that they were seen by night, the sources of light being the gas lamps and electric lights of the city. It was on the evening of Friday, November 11, after several days of thick foggy weather. The tiny fog particles seem to have gradually settled down in the still air, the fog steadily clearing the whole time. These globules, in spite of their contact with the damp ground, must have retained their spherical form almost perfectly; for they were able to throw back to the eye, with a rainbow deviation of approximately 42°, the light incident upon them.

In the neighbourhood of every lamp where the surface of the ground had been undisturbed by traffic the bows were seen as bright streaks, which shifted position with the observer. Any disturbance of the surface, a wheelmark, a footprint, a finger drawn across the pavement, obliterated the bright streak; and it was not seen on the surface of puddles.

1. There are two methods for expressing in determinant form the eliminant of three ternary quadrics. If u, v, w are the quadrics and J their Jacobian we may eliminate dialytically the ten quantities x3, y3, z3, y2zyz2, z2x, zx2, x2y, xy2, xyz from the ten cubics xu, yu, zu, xv, yv, zv, xw, yw, zw, J. This is in effect the process given by Sylvester in 1841. The other method is to eliminate dialytically the six quantities x2, y2, z2, yz, zx, xy from u, v, w and the three differentials of the Jacobian.

2. The eliminant of four quaternary quadrics may be found in determinant form by a process having points in common with each of these two methods. Multiplying each of the quadrics by the variables x, y, z, w in turn we get sixteen quaternary cubics, and we have also the four differentials of the Jacobian. Thus we have in all twenty quaternary cubics, or just sufficient to eliminate dialytically the twenty expressions x3, …, x2y, … xyz, … The result is a determinant of order 20, and of the correct degree, viz., 8, in the coefficient of each quadric.

It does not appear to be possible to extend this process so as to obtain, in determinant form, free from extraneous factors, the eliminant of a set of quadrics in more than four variables.

In continuation of former investigations of the alteration of compressibility of water, which is produced by dissolving various salts in it, I was led to imagine that some instructive results might be furnished by solutions such as those of sugar, whose bulk is nearly the sum of the bulks of their constituents:—for, in them, we might expect little change in compressibility from that of water itself; i.e. in accordance with my hypothetical formula, little change in the term regarded as representing the molecular pressure.

The following preliminary results have recently been obtained for me by Mr Shand, Nichol Foundationer, who employed the Fraser gun and the Amagat gauge procured for my “Challenger” work:—and a new set of piezometers of the same (Ford's) glass as that whose compressibility I had determined to be 0·0000026. These have been carefully gauged, but have not as yet been directly compared with those formerly employed.

The solutions experimented on were prepared, in Dr Crum Brown's Laboratory, by Mr W. W. Taylor, M.A., B.Sc., and contained respectively 5, 10, 15, 20 parts, by weight, of sugar to 100 of water. The temperature varied but slightly from 12°·4 C. during the whole course of the experiments.

When lecturing on Electrostatics at the Glasgow and West of Scotland Technical College, the author endeavoured to impress upon his students the identity of the laws of force for magnetism and electricity at rest. He thought it would further this idea if he could get pictures of electrostatic lines analogous to those obtained from magnets by means of iron filings, and an analysis of the difference of the conditions in the two cases led to the conclusion that there was no reason why this should not be done.

When we shower iron filings upon a glass plate, or a sheet of paper, under which a magnet is placed, unlike poles are induced at the ends of each filing, and it becomes for the time being a little compass needle. Since it is very short, the total action upon it will be a couple turning it into the direction of the force at its centre, unless it be close to the magnet, in which case there will be, in addition, a translation along a line of force. This latter effect causes the filings to crowd towards the edges of the magnet.

Now, in electrostatics we also have an inductive action, opposite charges being induced in the near and far ends of each particle as it comes into the field, but we have the important difference, that whereas the magnetic poles are attached to the filing and cannot be separated from it, the electric charges will leave a body if they have an opportunity of doing so.

§ 1. “Elastic solid” includes fluid and ether; except conceivable dynamics of the mutual action across the interface of the two mediums. Maxwell's electro-magnetic equations for a homogeneous non-conductor of electricity are identical with the equations of motion of an incompressible elastic solid, or with the equations expressing the rotational components of the motion of an elastic solid compressible or incompressible; but not so their application to a heterogeneous non-conductor or to the interface between two homogeneous non-conductors.

§ 2. The equations of equilibrium of a homogeneous elastic solid, under the influence of forces X, Y, Z, per unit volume, acting at any point (x, y, z) of the substance are given in Stokes' classical paper “On the Theories of the Internal Friction of Fluids in Motion, and of the Equilibrium and Motion of Elastic Solids,” p. 115, vol. i. of his Mathematical Papers; also in Thomson and Tait's Natural Philosophy [§ 698 (5) (6)].

The publication of Mr Nanson's paper on the glissette elimination problem (Proc. Roy. Soc., vol. xxii. p. 158) has led me to make a further investigation of this interesting point, with the result that the general eliminant is expressed in the form of a single symmetrical bordered determinant. I shall first give the solution for the case where the tracing-point is on the axis of the ellipse, and then extend it to the general case.

I. Case of the Tracing-point on the Elliptic Axis.

The equations for the centre of the generating ellipse as tracing-point are—

For a point in the line of the major axis, if r be the distance of the tracing-point from the centre, we have

whence, by squaring each side,

By addition,

When solutions of potassium persulphate and potassium iodide are mixed, iodine is liberated and potassium sulphate formed. The reaction takes place somewhat slowly in the cold, much more rapidly on warming. It has generally been supposed that the equation

2KI + K2S2O8 = I2 + 2K2SO4

completely represents the action which takes place, and although I have had occasion to try the reaction very many times since I first noted it as a reaction for persulphates, it was only about a year ago that I observed a further change.

A solution of potassium iodide was being boiled with ammonium persulphate (which is much more soluble than the potassium salt) in order to decompose it and drive off the iodine by sublimation. It was observed that the free iodine in the liquid seemed to disappear very rapidly, more so than corresponded to the amount volatilised, and that the final disappearance of the brown colour took place with unexpected suddenness. This pointed to the possible conversion of the iodine into some compound, probably by the oxidising action of the persulphate. This proved to be the case, because, on the cautious addition of sulphurous acid to the colourless solution, iodine again appeared; on standing, the liquid again became decolourised, and was again turned brown by sulphurous acid. These operations could be repeated so long as persulphate remained in the solution. The action takes place slowly in the cold, and at high temperatures there is considerable volatilisation of iodine; when a solution of potassium iodide was corked up with ammonium persulphate in a test tube at the ordinary temperature, the iodine which separated in the solid form gradually dissolved on further standing, and had entirely disappeared in the course of several days.

I am very glad to embrace the opportunity of addressing the Fellows of the Royal Society of Edinburgh, many of whose members have contributed much to our knowledge of Oceanography. For instance, your President, Lord Kelvin, besides his researches on the tides, is well known to practical seamen from his excellent compasses and sounding machines; your Secretary, Prof. Tait, is well known from his researches on the pressure errors of deep-sea thermometers; Dr Alexander Buchan has a world-wide reputation in the department of oceanic meteorology; the late Prof. Dittmar was a great authority on the chemistry of sea water. It is enough to say that Sir Wyville Thomson, Mr J. Y. Buchanan, and Sir John Murray, were members of the “Challenger” Expedition, which has given the world such valuable information about the depths of the sea.

Of course, it is not with the intention of giving to such scientific authorities a lesson that I address the Society, but if you represent scientists, I represent the seamen, and it is useful from time to time to have a talk between these two classes of men. Every scientific study should be started by the scientist, but the sooner they can associate ordinary practical men with the work the better it will be. We practical seamen are more numerous than scientists; we constantly navigate the sea, and we have more opportunities of making contributions to science than they have. Certainly, they can make their observations in a more exact way than we can, but the laws of nature—particularly those concerning Oceanography—are so imperfectly known, that there is very much to be done even by the rough hands of the ordinary seamen.

The following is a brief account of the principal points of interest resulting from the survey lately made to find a suitable route for a cable to be laid from Bermuda to Turk's Islands and from thence to Jamaica.

As engineers to the Direct West India Cable Co., my firm, Messrs Clark, Forde, & Taylor, strongly advised that a more than usually complete set of soundings should be taken along the route over which it was proposed that the cable should be laid, our principal reason being that, as the Bermuda group has been raised almost vertically from the bed of the Atlantic by volcanic action, it was impossible, with the limited information available, to say whether similar action might not have taken place elsewhere on the proposed route for the new cable.

The task of carrying out this survey was undertaken by the Telegraph Construction and Maintenance Co. with their s.s. ‘Britannia,’ which was fitted up with all necessary apparatus, and was also well adapted for the work.

Sounding operations were commenced off Grand Turk Island on the 4th November 1897, and completed on the 2nd December following at Bermuda, during which time 182 soundings were taken, apart from the profile soundings taken off Grand Turk. The laying of the cable was commenced at Bermuda on the 5th January 1898, and completed on the 18th of the same month.

As a result of this work the following suggest themselves as points worthy of notice:—

(1) The even slope of the ground from the neighbourhood of Bermuda to the deepest water found in Lat. 24° 50′ N., no indication being discovered of any upheaval similar to that which has produced the Bermudas with their adjoining shoals.

The number of experiments that have been made on the Röntgen rays is already very large, and much has been learned from them regarding the properties of these rays. There has been, however, as yet, but little done to determine their energy. Indeed, the only accurate work which has been carried out in this direction is that of E. Dorn (Wied. Ann., Bd. 63, p. 160), who measured the heat produced by Röntgen rays falling on a metal plate. This investigation is, however, not complete, as Dorn did not determine the number per second or the duration of the discharges of the Röntgen rays.

On the suggestion of Professor E. Wiedemann (Erlangen), in whose laboratory I have been working, I have tried to investigate this subject; and, although I have not had time to complete my experiments, as I now find I have to return to India much earlier than I expected, I shall describe the methods adopted and the results obtained, in the hope that they may not be without interest.

I first sought to determine in absolute measure the quantity of light produced on a screen of barium platino-cyanide by Röntgen rays. For this purpose I used an optical bank with the barium platino-cyanide screen and the Röntgen lamp at one end, the source of light which I used as a standard of comparison at the other end, and between them a movable photometer. The arrangement of the apparatus is shown in the following diagram.

In vol. vi. of the Zeitschrift für Physikalische Chemie (1890), pp. 16–36, Nernst demonstrates the relation between the osmotic pressure of a given solution of N in A and the difference of concentration of two solutions of A in B, the one made by shaking up B with A and the other by shaking up B with the solution of N in A; where A and B are two liquids miscible with each other, but not in all proportions, as, for instance, water and ether, and N a substance soluble in A but not in B. Immediately after this paper, Nernst describes (l.c., pp. 37–40) an osmotic experiment in which the “semipermeable membrane” is a layer of the liquid B held in its place by capillarity. Through this layer no N can pass, because N is insoluble in B, but A will pass from what we may call the A side, on account of the concentration gradient, the layer of B containing more A dissolved in it on the A side than on the solution side. At the same time a pressure is developed on the solution side equal to the osmotic pressure of the solution of N.

So far as the diffusion of A through the layer of B from the A side to the solution side is concerned, Nernst's experiment can be shown without fixing the layer of B. In the form exhibited to the Society, A is water, B phenol, and N calcium nitrate.

(1) As is well known, the first rule given for ascertaining the sign of a single term of a determinant was made known by Cramer in his Introduction à l'Analyse des Lignes Courbes algèbriques, published at Geneva in 1750. On page 658 he says—

“On donne a ces termes les signes + ou −, selon la Règle suivante. Quand un exposant est suivi dans le même terme, médiatement ou immediatement, d'un exposant plus petit que lui, j'apellerai cela un derangement. Qu'on compte, pour chaque terme, le nombre des dérangements: s'il est pair ou nul, le terme aura le signe +; s'il est impair, le terme aura le signe −. Par ex. dans le terme Z1Y2V3 il n'y a aucun dérangement: ce terme aura done le signe +. Le terme Z3Y1X2 a aussi le signe +, parce qu'il a deux derangements, 3 avant 1 & 3 avant 2. Mais le terme Z3Y2X1 qui a trois derangements, 3 avant 2, 3 avant 1, & 2 avant 1, aura le signe −.”

According to Cramer, therefore, If δ be the number of derangements in the permutation corresponding to any term, the sign of the term is (−)δ.

Instead of the word “dérangement,” Gergonne in 1813 used “inversion” in speaking of Cramer's rule: Cauchy did the same in 1841: and, consequently, the latter term or “inversion of order” has come into pretty general use. “Inverted-pair” is probably a still better expression.‡

In the article “Tides” in the Encyclopœdia Britannica, Professor George Darwin, quoting Sir George B. Airy, remarks that the tides of the North Sea present a very remarkable peculiarity. Along the eastern coast of England as far as the mouth of the Thames, the tide-wave, coming from the Atlantic round the Orkney Islands, flows towards the south. Thus, on a certain day, it is high water in the Moray Firth at 11 a.m., at Berwick at 2 p.m., at Flamborough Head at 5 p.m., and so on to the entrance to the Thames. Thus, on the day supposed, it will be high water off the Thames at 11 p.m., the tide having travelled in twelve hours from the Moray Firth.

It is further stated that the North Sea is considerably deeper on the English side than on the German side; so much so, that the tide-wave coming from the north runs into a deep bay of deep water, bounded on the west side by the Scottish and English coasts as far as Newcastle, and on the east side by the great Dogger Bank. As far as the latitude of Hull, the English side is still the deep one; and though a species of channel through the shoal there allows an opening to the east, yet immediately on the south of it is the Wells Bank, which again contracts the deep channel to the English side.

It is not stated here that the deeper water of the North Sea close to the Scottish and English coasts determines the course of the southward tidal flow to be close to these coasts; but for that course taken, no other cause is suggested.

An examination is made of a series of early stages in the development of the shoulder-girdle in the common Phalanger (Trichosurus vulpecula), and of early stages in the development of that of the Ring-tailed Phalanger (Pseudochirus peregrinus), and of the Rock-wallaby (Petrogale penicillata). Though in their main features there is much similarity between the girdles of the different genera, there are certain peculiar features in each.

In a Trichosurus fœtus of 8·5 mm., though the main part of the scapula is chondrified, together with the glenoid portion of the coracoid and a portion of the acromion, the rest of the arch, the sternum, and the greater part of the clavicle, are still mesenchymatous. The coracoid can be traced downwards as a fan-like expansion, which meets the first rib and the sternum. Between the coracoid and the clavicle a feebly-developed thin sheet apparently represents the precoracoid (epicoracoid). The clavicle is partly ossified at its upper part, and it is very manifest that there is no cartilaginous basis.

In the Trichosurus fœtus, at birth (14 mm.), the scapula is well developed. The acromion is a large process which springs from the anterior border of the blade. The spine is not yet formed, though its basis can be distinctly traced as a membranous structure stretching from the acromion upwards along the outer side of the scapula in its anterior third. There is no cartilaginous basis for this part of the spine. The coracoid is of large size, and its lower part, which is somewhat bulbous, articulates with the sternum and with the first rib.

In the year 1847, as one of the results of an investigation to determine the part played by the mass action of water in chemical reactions, H. Rose (Pogg. Ann., lxxxii. 545) showed that an acid sulphate in aqueous solution is progressively decomposed into free acid and neutral sulphate by increasing quantities of water. His observations were confirmed and extended by the thermo-chemical researches of Thomsen (Pogg. Ann. 1869, cxxxviii. 72), and Berthelot (Ann. Chim. Phys., 1873, xxix. 433), who indicated more exactly the course and extent of this decomposition, and from whose work it is known that in the solution of an acid sulphate there exist free sulphuric acid, neutral sulphate and acid sulphate. Finally Ostwald, in his first memoir on chemical affinity (Jour, prakt. Cliem. 1879, xix. 483), showed how to determine the magnitude of this decomposition, and was able to approximately measure the quantity of free acid in solutions of the acid sulphates at different dilutions. In a later paper (ibid., 1880, xxii. 305), Ostwald investigated the question of the influence of water on the action between sulphuric acid and a neutral sulphate. He measured the changes of volume which occurred when solutions of sulphuric acid and sodium sulphate were mixed in varying proportions, and at different dilutions, and obtained results in agreement with his previous work. He also pointed out, that if the mutual action between the acid and neutral sulphate obeyed the general law of mass action, it should follow that, for example, one molecule sodium sulphate plus three molecules sulphuric acid, give the same quantity of acid sulphate as one molecule sulphuric acid, and three molecules sodium sulphate, the volume of the mixture being the same in each case.

§ 1. For a perfectly definite mechanical representation of Sellmeier's theory, imagine for each molecule of sodium-vapour a spherical hollow in ether, lined with a thin rigid spherical shell, of mass equal to the mass of homogeneous ether which would fill the hollow. This rigid lining of the hollow we shall call the sheath of the molecule, or briefly the sheath. Within this put two rigid spherical shells, one inside the other, each movable and each repelled from the sheath with forces, or distribution of force, such that the centre of each is attracted towards the centre of the hollow with a force varying directly as the distance. These suppositions merely put two of Sellmeier's single-atom vibrators into one sheath.

§ 2. Imagine now a vast number of these diatomic molecules, equal and similar in every respect, to be distributed homogeneously through all the ether which we have to consider as containing sodium-vapour. In the first place, let the density of the vapour be so small that the distance between nearest centres is great in comparison with the diameter of each molecule. And in the first place also, let us consider light whose wave-length, is very large in comparison with the distance from centre to centre of nearest molecules. Subject to these conditions we have (Sellmeier's formula)

where m, m denote the ratios of the sums of the masses of one and the other of the movable shells of the diatomic molecules in any large volume of ether, to the mass of undisturbed ether filling the same volume; k, k the periods of vibration of one and the other of the two movable shells of one molecule, on the supposition that the sheath is held fixed; ve the velocity of light in pure undis turbed ether; vs the velocity of light of period τ in the sodium-vapour.

The phenomena to which it is desired to call attention were first observed in the course of an investigation into the electrolytic methods of determining nickel, carried out in connection with the work of the British Association Committee on the Electrolytic Methods of Quantitative Analysis. The results of the investigation are given in the Report of last year's meeting (Bristol, 1898, p. 300), but the present matter is not referred to there, as it was first noticed only during the preparation of the report.

The apparatus employed for most of the determinations consisted of the usual Classen form of cathode basins, with disc anodes. The current was drawn from storage batteries at 12 volts, variable resistances being used in the circuits. A Davies' ammeter could be introduced easily into any circuit at will, and a Davies' voltmeter was also used to indicate the difference of potential at the electrodes. These measuring instruments were so arranged and connected that the operation which introduced the ammeter into a circuit simultaneously brought the voltmeter into action; hence it happened that voltmeter readings were always taken conjointly with ammeter ones, although attention was paid almost entirely to the effects of varying the current density, and to any changes which might occur during the course of the experiment, with no special consideration of the potential involved.

For the determination of quantities of nickel varying between 0·l and 0·5 gram, the conditions adopted as most generally suitable when employing the usual method were those given below:—

Volume of solution, about 135 c.c. (giving about 100 sq. cm. of cathode area), containing 5 grams each of ammonium sulphate and of ammonia; current density at cathode (amp. per sq. decim.), 0·6–0·7; ordinary temperature; time necessary, about 4 hours.

(1) As is well known, the simplest form of alternant is

and the problem of multiplying it by any symmetric function of a, b, c, d, … has been in a manner fully solved.

(2) When the symmetric function is linear in each of the variables—that is to say, when it takes any of the forms Σa, Σab, Σabc, ….—the result is an alternant got from the multiplicand by increasing the last index, the last two indices, the last three indices, …. respectively by 1. Thus, writing for shortness' sake five variables only, we have

This was first established in 1825 by Schweins in his Theorie der Differenzen und Differentiale, p. 378; but it is also barely possible that it was known to Prony in 1795 (see Journ. de l'Ec. Polyt., i. pp. 264, 265), and Cauchy in 1812 (see Journ. de l'Ec. Polyt., x. pp. 49, 50).

(3) When the symmetric function is non-linear, the result takes the form not of one alternant, but of an aggregate of alternants. These cannot be so readily specified, but the mode of obtaining them can be made clear without any difficulty. Let us take the case of the function Σa3b the multiplicand being ∣a0b1c2d3∣.

1. The fact that the eliminant of the set of five equations

is, as shown by Professor Schoute, the persymmetric determinant

implies a linear relation connecting every three consecutive s's. As a matter of fact it is not difficult to see that we have

and from these, by elimination of λ1λ2, λ1+λ2, we obtain the result just mentioned.

Three years ago I called the attention of the Society to the following theorem:—

The resultant of two pure strains is a homogeneous strain which leaves three directions unchanged; and conversely.

[It will be shown below that any strain which has three real roots can also be looked on (in an infinite number of ways) as the resultant of two others which have the same property.]

As I was anxious to introduce this proposition in my advanced class, where I was not justified in employing the extremely simple quaternion proof, I gave a number of different modes of demonstration; of which the most elementary was geometrical, and was based upon the almost obvious fact that

If there be two concentric ellipsoids, determinate in form and position, one of which remains of constant magnitude, while the other may swell or contract without limit; there are three stages at which they touch one another.

[These are, of course, (1) and (2) when one is just wholly inside or just wholly outside the other (that is when their closed curves of intersection shrink into points), and (3) when their curves of intersection intersect one another. The whole matter may obviously be simplified by first inflicting a pure strain on the two ellipsoids, such as to make one of them into a sphere, next considering their conditions of touching, and finally inflicting the reciprocal strain.]

But the normal at any point of an ellipsoid is the direction into which the radius-vector of that point is turned by a pure strain; so that for any two pure strains there are three directions which they alter alike. (These form, of course, the system of conjugate diameters common to the two ellipsoids.) This is the fundamental proposition of the paper referred to, and the theorem follows from it directly.