The following problem was sent me by Mr Kirkman, F.R.S.; its solution illustrates the Calculus of Relationship, Proc. Roy. Soc. Edinb., vol. xi. pp, 5 and 162:—

Two boys, Smith and Jones, of the same age, are each the nephew of the other; how many legal solutions?

The statements are that Smith is the nephew of Jones, and Jones the nephew of Smith. Let c denote child, then denotes parent, and the equations are

Hence .

In but a few cases are solutions of equations of the type

known, and even in the simplest cases the most general of the known solutions suffers from the disadvantage of being in a symbolic form. The equations to be studied in the present paper are those which can be derived from

by a change in the independent variable. This equation is the one which most naturally claims the attention of the investigator, after the equation of wave-motion

has been disposed of.

At long intervals notices of circular crystals have appeared before this Society. In 1853 Sir David Brewster read a paper on the subject, which followed one by Mr Fox Talbot in 1836, and which again had been preceded by one from Sir David Brewster about twenty years before. It is not easy to account for these long intervals, unless they may be attributed to difficulty and uncertainty in manipulation, for except in very few instances the crystals observed by Sir David Brewster are of microscopic size, and, he remarks, require the perfection of optical appliances for their observation, and naturally so when crystals of the 200th of an inch in diameter are looked upon as of respectable size.

The first obvious difficulty which presents itself, in trying to derive Clerk-Maxwell's equations from those of the elastic-solid theory, appears in the fact that the latter, being linear, do not impose any relations among simultaneous disturbances. Thus, for instance, they indicate no reason for the associated disturbances which, in Maxwell's theory, constitute a ray of polarised light. Hence it appears that we must look on the vectors of electric and magnetic force, if they are to be accounted for on ordinary dynamical principles, as being necessary concomitants, qualities, or characteristics of one and the same vector-disturbance of the ether, and not themselves primarily disturbances. From this point of view the disturbance, in itself, does not correspond to light, and may perhaps not affect any of our senses. And the very form of the elastic equation at once suggests any number of sets of two concomitants of the desired nature, which are found to be related to one another in the way required by Maxwell's equations.

The extinction of the Tasmanian aboriginal in 1876 closed, for all practical purposes, the further scientific study of this ancient and highly interesting race, and it appeared almost certain that our knowledge of this people would remain dependent on the earlier works of those who were fortunate enough to have studied them during life, and on the few remains housed in such fortunate centres as London, Paris, Edinburgh, Oxford, and Cambridge.

1. The morphology and anatomy of Trichomanes aphlebioides Christ, a species endemic to New Guinea, is shortly described.

2. The most outstanding feature of the species is the presence of aphlebioid leaves. These latter are shown to be the modified first fronds of the axillary branches.

3. The anatomy of the alphebioid leaves is described. They show great reduction in relation to the humid conditions of the high forests in which the plant grows.

4. The morphology of aphlebiæ and aphlebioid leaves is briefly discussed.

5. No experimental evidence is available with regard to the function of the aphlebioid leaves. Mention is made, however, of the various suggestions which have been put forward in relation to this question.

The effect of the friction of the crushing plates upon the yield point of short compression specimens has been investigated.

With plates harder than the material under test, the end friction causes an increase in the apparent yield point. This increase is calculated as 18 per cent. for wrought iron and mild steel, 20 per cent. for cast iron, and 50 to 200 per cent. for stones, bricks, and concrete. These figures, except the first, may apply almost equally well to the crushing strength, but they require experimental support. The corresponding inclinations of the surfaces of shearing are 37° for wrought iron and steel, 36° for cast iron, and 27° to 15° for stones, etc. The first value is seldom obtained, but the others agree fairly well with average experimental results.

When the crushing plates are of material softer than that under test, the lateral flow of the former diminishes the apparent strength of the specimen. With stones crushed between lead plates the strength is calculated as 0·35 to 0·15 of that obtained when iron or steel plates are employed. Experiments give 0·65 to 0·43 as the value of this ratio, but the specimens do not rupture by shearing in the manner contemplated.

The total crushing load of a short specimen of cast iron is increased by diminishing the length of the piece, but the crushing stress per unit of area is simultaneously decreased.

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.

Two earthquake shocks have lately occurred in Argyleshire of so decided a character that a description of their effects, as observed at four of the Lighthouse stations on the west coast of Scotland, will, it is thought, be interesting to the Society.

The first shock occurred on the 11th March, and was observed at the Lighthouse station of Hynish in the island of Tyree, and at Sound of Mull, near Tobermory, the distance between the two places being about 34 statute miles.

The periodic variations of the temperature of water have been studied for some months at the Scottish Marine Station at Granton. The only tidal basin which has been considered as yet is the haven formed by the irruption of the sea into Granton Quarry. It has an area of about 7 acres; the tidal entrance is on the west side, and is so situated that no water can enter until about half tide; then it comes in as a very rapid stream for about three-quarters of an hour, when the rate falls off, and near high water it is the same as that of the rising tide on the shore. The ebb is gradual until the tide has narrowed the channel by uncovering the sandbanks which lie on each side of the entrance, then it is rapid for about an hour and a half, after which the water runs out ext'remely slowly, and does not absolutely cease until the tide begins to re-enter; but for from four to five hours the water-level inside is practically unaltered. The depth of the quarry at low water varies from 5 to 8 fathoms in the parts where observations were made, and at high water the depth is slightly more than 1 fathom greater. The bottom shelves off abruptly from the bar at the entrance.

It became evident during the War that some men who are organically sound are nevertheless incapable of supporting heavy exertion; and among the older groups that were called to the Colours in 1918 the proportion of such individuals was considerably more. The effect of forcing these men to carry out the duties falling to the lot of an “A” recruit was in many cases harmful to them, and in extreme instances resulted in permanent disability and sometimes in death. The Army Medical Department therefore sought to obtain a method of physical examination, which would supplement the ordinary medical examination of a recruit, and which could be applied to men who proved to be unserviceable material in the hands of the drill sergeant—a test which would, in fact, enable the malingerers to be distinguished from those who were truly incapable of sustained labour.

In 1877, shortly after the disastrous colliery explosion in the Blantyre pit, in which hundreds of lives were lost, Mr James Young, F.R.S., of Kelly, described to me an instrument which he had thought of for determining what is the amount of fire-damp in any part of a mine. This was the first thing which directed my attention to the subject, and I very soon saw that there was a principle in acoustics which might be most admirably adapted to the end in view, viz., to determine the quantity of fire-damp (or marsh gas) by the diminution in density of the mixed air and gas (for marsh gas is only about half the density of air). Mr Young and myself tested the principle the next day, and found it to be one of extreme delicacy.

In our last communication (read July 2, 1894) we gave the broad results for three nickel tubes of various bores (see vol. XX. p. 296). In these experiments the screw stopper through which the capillary glass tube passed was made of brass. This brass stopper was used because of the comparative difficulty of working nickel. We were led, in the course of some tentative experiments, to use this brass stopper with one of the steel tubes. Now, in the earlier experiments with the iron and steel tubes, we had used an iron stopper. We were hardly prepared, however, to find that the mere substitution of a brass for an iron stopper should produce such a marked effect, not only on the amount of change of volume in given magnetic fields, but also on the manner in which the volume change varied as the field increased or decreased steadily.

The author remarks that the existence of Annelida during the Palæozoic formations is manifested in two conditions. In the one, we have the shelly envelope which invests the order Tubicola, in the form of Seapolites; and in the other, the tracks of the orders Abranchia and Dorsi-branchiata are found impressed on deposits which were, at one time, in a sufficiently soft state to receive the impressions of the wanderings of these animals.

Among the strata which have hitherto afforded annelid tracks, those which, in the county of Clare, represent a portion of the equivalents of the Millstone Grit, contain such tracks, in their most perfect state of preservation in great abundance; and these strata also furnish evidence concerning the circumstances which prevailed during their deposition.

A series of reflex adjustments of balance disturbed round the anteroposterior and horizontal axes are described in the domestic duck.

In previous papers by Desai and collaborators (Desai and Borkar, 1933 a; Desai and Desai, 1933 b; Mankodi, Barve, and Desai, 1936 a; and Joshi, Barve, and Desai, 1936 b) it has been shown that the changes which are produced in the charge on the colloidal particles during dialysis of ferric hydroxide, thorium hydroxide, prussian blue, and arsenious sulphide sols are not so simple as is usually supposed, and that data for viscosity and stability towards electrolytes do not necessarily give direct information with regard to the charge on the colloidal particles. In the present paper measurements of cataphoretic speed in the presence and absence of electrolytes, stability towards electrolytes, and conductivity of colloidal solutions of gold and vanadium pentoxide which have been dialysed, diluted, allowed to age, and exposed to sunlight to different extents, have been presented and the interpretations given in the previous papers amplified.

Ever since the remarkable discovery by Graham in 1866 that palladium possesses the power of occluding hydrogen gas in large quantity, various views have from time to time been put forward to explain the true nature of the phenomenon. Whilst some observers regard hydrogenised palladium as an alloy or solid solution, others again consider it to be, or at least to contain, a definite chemical compound or hydride. Those who hold the latter opinion, however, are not agreed as to which compound or hydride is formed, as is proved by the fact that different formulae have been ascribed to it. Graham himself recognised the possibility that a definite chemical compound might be formed, for he says (Researches, 287) that in fully-charged palladium there exists one equivalent of palladium to 0·772 equivalent of hydrogen, or an approximation to single equivalents which would be represented by the formula PdH. His opinion was, nevertheless, opposed to the idea of such a definite chemical combination, one of his chief objections being that no visible change is occasioned to metallic palladium by its association with hydrogen. He regarded the product simply as an alloy of the volatile metal hydrogenium, in which the volatility of the one element is restrained by its union with the other, and which owes its metallic aspect equally to both constituents.

Considerations of a purely chemical character have up to the present time proved insufficient to decide which of these views is correct.

The problem in question is to determine the equation of the curve traced out by any point of an elliptic or hyperbolic disc which touches two fixed rectangular axes. Mechanically constructed figures of different forms of the curve have been given by Tait, who also showed that the same glissette can be traced either by means of an ellipse or a hyperbola. If p, q are the coordinates of the tracing point referred to the axes of the disc as axes of coordinates, the glissette is clearly the θ eliminant of

and Cayley stated that it would be found to be of order 8 in x, y. The actual elimination was first performed by Muir, who obtained in the first instance an equation of order 10. On dividing by an extraneous quadratic factor a lengthy equation of order 8 was obtained; and subsequently Lord M'Laren verified the accuracy of the terms of highest order in this equation.

The paper contains some curious applications of the principle of dissipation to the conduction of heat, the connection of heat and electricity, thermo-electric currents, the electric convection of heat, &c. But in this abstract we confine ourselves to one very simple case of the conduction of heat, as the hypothesis on which it is investigated is fundamentally assumed in all the other applications.