This is a direct continuation of an earlier communication by the author, on “The Revised Prepared System of the Quadratic Complex,” which appeared in vol. lvi (1936) of the Society's Proceedings, pp. 38–49. It will be convenient to utilize the same notation and references to earlier works I, II and III as before, with IV to denote the 1936 paper. A complete system of concomitants for a quadratic complex is here worked out; omitting polar forms, it consists of 210 forms. The sign ≡ in a relation A ≡ B is here used to mean “may be replaced by”: thus A is equal to a non-zero numerical multiple of B plus reducible terms.

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 elliptic cylinder functions, or solutions of the Mathieu equation

that have the period π or 2π, are of four distinct species. These are denoted respectively by ce2n(x, θ), se2n+1 (x, θ), ce2n+1 (x, θ), se2n+2(x, θ), where n indicates the number of zeros in the open interval o < x < ½π. The object of this paper is to show that a function of any type can be expressed in terms of functions of other types, or of their derivatives. This provides the analogies to such trigonometrical relations as

but the single terms on the right-hand side of this and similar relations are replaced, in the results to be proved, by infinite series. These results are quite distinct from Whittaker's recurrence-relations (Journ. London Math. Soc., vol. iv, 1929, pp. 88–96); in particular, they do not involve the non-periodic second solutions.

The state of knowledge of fast neutron polarisation due to elastic scattering is reviewed in relation to the optical model of the process. There is clearly a substantial lack of polarisation data at neutron energies and concerning nuclei appropriate to the optical model description. The situation for the extremes of small and of large angle scattering is also considered.

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.

In cases of cleft palate with alveolar fissure, the maxillary bones are, during infancy, not only ununited, but, in general, if not always, more widely separated from one another than in the natural condition. This has been frequently observed in such cases as come under the care of the surgeon, although little attention appears to have been bestowed upon the fact beyond its mere casual mention. Measurements, however, have been lately made by Dr Engel, showing that the difference between certain fixed points—such as the two infra-orbital foramina, the nasal processes of the upper jaw, &c, is very well marked, when the distance is measured in a healthy new-born child, compared with one having a cleft palate.

It has been mentioned by Professor Voigt of Göttingen in his newly published book and by Professor Zeeman of Amsterdam in the Physikalische Zeitschrift, that I have found examples of strongly marked dissymmetry in studying the Zeeman Effect in tungsten and molybdenum. Professor Zeeman has also discovered and published such cases of dissymmetry in other elements. Not only have many examples of normal dissymmetry been found, but almost as many cases of abnormal dissymmetry. To explain those terms, normal and abnormal, let us consider that the single spectrum line is broken up, when the light is in the magnetic field, into the three components, 1, 2, 3, where the numbers begin from the component which has the shortest wave-length. In the normal dissymmetrical triplet the middle component is nearer the component on the red side than that on the violet one, i.e. for the normal type the interval 1–2 is greater than the interval 2–3, but in the abnormal dissymmetrical triplet 2 is nearer to 1 than to 3. These observations of Professor Zeeman and myself, which were made at the same time in the Universities of Amsterdam and Göttingen, having been communicated to Professor Voigt, he wrote and published in the above-mentioned book an extension to his and Professor H. A. Lorentz's theories of the Zeeman Effect. In his original theory Professor Voigt had shown that, considering the electrons as uncoupled, cases of normal dissymmetry might arise among the Zeeman triplets, this dissymmetry being accompanied by a greater intensity of the red component than the violet one. He pointed out also that the ‘absolute’ dissymmetry or the difference between the absolute displacements of the red and violet components should be independent of the magnetic field strength used to produce the Zeeman Effect. To explain the large numbers of complicated types of Zeeman Effect which have been found —in the study of the Zeeman Effect in tungsten I discovered lines with no fewer than 17 to 19 components, the largest numbers hitherto found—Professors Voigt and Lorentz made use in their theories of couplings between electrons of the same vibration frequencies.

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

While the occurrence of the condition termed Cyclopia cannot be considered as altogether a rare event, the obscurity in which its pathology is still involved makes it desirable that a full description of every case should be put on record. The specimen in my possession was that of a well-formed female embryo which had apparently reached the seventh month. With the exception of the malformation to be specially considered, there was no abnormality either in its external appearances, or in the structure and disposition of any of its viscera.

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.

In a paper communicated to this Society last session, I pointed out that the proof of Euc. I. 5, given by Mr Mill, is unsound; endeavouring, at the same time, to show that this is no mere accident, but that it is impossible to give a mathematically correct analysis of the processes of Synthetic Geometry on any theory that holds figures to be merely illustrative, and does not admit that intuition in the Kantian sense—i.e., actual looking at a single engraved or imaginary figure—may be a necessary and sufficient step in a demonstration perfectly general.

In taking cranial capacity as a ground of comparison between the Anthropoids and Man, he gave some measurements of skulls contained in the Museum. The capacity of the cranial cavity in the Gorilla skeleton belonging to the Museum is 29 cubic inches. Considerably larger gorilla skulls, however, have been measured. The smallest healthy adult human skulls in the Museum are those of two Andaman Islanders. Of these, one has a cranial capacity of 75·5 cubic inches, and the other a capacity of 72 cubic inches. Next to these comes a Peruvian skull; its cranial cavity measures 76 cubic inches. The smallest healthy adult human skull measured by Morton was also that of a Peruvian, whose cranial capacity measured only 58 cubic inches.

The facts to be explained were observed by Messrs Gore and Barrett, and were described by the latter gentleman in the “Philosophical Magazine” for 1873.

The experiments are performed on an iron or steel wire of no great thickness. When this is heated to an intense white heat and allowed to cool, the following facts appear at the instant it has cooled down to a dull-red heat:—

1. The wire expands for an instant, and then continues its normal contraction.

2. The glow from the wire is at the same instant seen to increase.

The homogeneous linear equation of the second order may be solved by means of a, continued fraction whose elements are coefficients in the equation. The two dimensional continued fraction is generalised to m dimensions and yields the solution of the equation of the mth order.

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 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 ages of the various tribes and sub-tribes of Compositæ have been calculated in doubling periods, according to Yule's formulæ; these are known briefly as Dp-ages (Small and Johnston, 1937; Udny Yule, 1924). Yule advises caution in interpreting Dp-ages in terms of geological time, but if his conceptions are to be applied to realities there should be some method of comparing Dp-ages with actual ages. Further, in the case of Compositæ, an almost complete parallel seriation has been found for Dp-ages and order of origin of both tribes and sub-tribes (Small and Johnston, 1937). Since the order of origin has been provisionally correlated with geological time (Small, 1919) it is possible to compare the calculated Dp-ages with suggested points of origin in geological time. This has been done by Small and Johnston (1937, Tables, VI, VII).

The angular rotation of the plane of polarization of light-rays in their passage through quartz is a function of the wave-length, and is roughly represented by the formula

where A is a constant depending on the quartz. This formula is only approximate, however, and one object of the experiments described below was to ascertain how closely the rotation might be represented by three terms of the equation

Some time ago the author commenced an investigation of the Zeeman effect in the spark spectrum of certain rare elements. The research promised results of considerable interest, as the properties and relationships of these substances are as yet very imperfectly known, and certain of the elements show indication of high magnetic quality. The following elements were examined: uranium, rubidium, gadolinium, samarium, scandium, dysprosium, neo-ytterbium, and lutecium. The author is greatly indebted to Sir William Crookes for his kindness in supplying the scandium, and to Professor Urbain of the University of Paris for the samples of the three last-mentioned elements.