I was recently led to consider this subject by Professor A. Dickson, who showed me some of his beautifully-mounted specimens, and explained to me the method he employs for the determination of the divergence, and of the successive leaves of the fundamental spiral or spirals. He referred me to two terribly elaborate papers by Bravais, and I have since met with another of a similar character by Naumann. These papers certainly cannot be supposed to present the subject from the simplest point of view.

§ 1. Object of Paper.—The chief object of the present paper is to establish the following theorem:

The functions of Lamé [that is to say, the doubly-periodic solutions of the differential equation

are the solutions of the homogeneous integral-equation

where η denotes an arbitrary constant.

This paper concludes the study of fitting polynomials by Least Squares, treated in two previous papers. The problem being concerned with the minimum of a positive definite quadratic form, it makes for conciseness to use matrix notation. We shall therefore adopt the following conventions :—

The n values of the variable x, of the data u0, u1, …, un−1, of certain polynomials qr(x) entering into the solution, and so on, will be regarded compositely as vectors. They will be imagined as having their components or elements disposed in column array, but when written in full will be written horizontally, to save space, enclosed by curled brackets. Row vectors, when written out in full, will be enclosed by square brackets. In the shorter notation we shall write, for example, u, x for column vectors, u′, x′ for the row vectors obtained by transposition. The vectors occurring in the problem will be the following:—

On the Mechanical Action of Radiant Heat or Light.

It is assumed in this communication that the undulatory theory of radiant heat and light, according to which light is merely radiant heat, of which the vibrations are performed in periods between certain limits of duration, is true. “The chemical rays,” beyond the violet end of the spectrum, consist of undulations of which the full vibrations are executed in periods shorter than those of the extreme visible violet light, or than about the eight hundred million millionth

During the first seven months of the year 1908 I had the good fortune to be living at Invergarry in Inverness-shire, with sufficient time at my disposal to make temperature observations in Loch Garry, and, through the generosity of Sir John Murray and Mr Laurence Pullar, the funds of the Lake Survey (Pullar Trust) were put at my disposal to defray the expenses of observation. I was further fortunate in securing the services of Mr Wm. Macdonald, Mount Pleasant, Fort Augustus, who was previously boatman to the Lake Survey on Loch Ness, and who proved himself a most painstaking and eager observer, and by whom by far the greater number of the actual observations were made.

The view that a peculiar likeness exists between the northern and southern extra-tropical faunas, and particularly between those of the Arctic and Antarctic regions, was suggested by Théel in discussing the remarkable deep-sea group of the Elasipoda, whose discovery we owe to the Challenger Expedition. A somewhat similar view is hinted at or referred to more than once in other Reports of the same Expedition. It was afterwards stated in an ampler way by Pfeffer (Versuch über die Erdgeschichtliche Entwickelung der jetzigen Verbreitungsverhältnisse unserer Thierwelt, 1891), and has of late been dealt with in great detail, and in relation to the antecedent causes that might have led to such a phenomenon, by Sir John Murray. On the other hand, Dr Ortmann, considering the hypothesis from the point of view of our knowledge of the distribution of Crustacea, has rejected it entirely (“Uber Bipolarität in der Verbreitung mariner Thiere,” Zool. Jahrb., 1896; cf. also “Marine Organismen und ihre Existenzbedingungen,” ib., 1897), and Dr Chun, dealing with the pelagic fauna (“Die Beziehungen zwischen dem arktischen und antarktischen Plankton,” Stuttgart, 1897), while showing how in truth a certain small number of forms are common to far northern and far southern seas, holds that the facts are sufficiently accounted for by the continuous distribution or gradual intermixture of forms in the depths of the intervening oceans under present conditions, without our needing to have recourse to an explanation of the phenomenon in the different conditions of a former age.

Vertebraria is one of the commonest and most characteristic fossils in those Upper Carboniferous and Permian strata in which Glossopteris is the dominant fossil plant. It has been found in the form of coaly incrustations, impressions, and casts, and accounts of these various forms have been given by Arber and Seward. In view of our very inadequate knowledge of Vertebraria and Glossopteris any information about them is of importance. The genus Vertebraria Royle was founded on impressions of plant axes, some of which, from the evidence of the attachment of roots, must be judged to be roots or rhizomes. Vertebraria axes are usually found lying in the plane of the bedding of the matrix (Pl. I, fig. 2) and less frequently at right angles to the plane of bedding (PL I, fig. 3).

Coefficients of inbreeding have been calculated for 676 stallions and mares, with the object of studying the part played by inbreeding in the development of the Clydesdale breed of horses. Wright's coefficient of inbreeding was used in this study.

It is found that during the early history of the breed very little inbreeding had been practised, and it is suggested that this may be due to the fact that the foundation stock contained many recessive factors for undesirable characters. The first marked rise in the average percentage of inbreeding for the breed occurs during the period 1880–1890, and from that time onwards there is a gradual but constant increase in the breed average, reaching its highest level at 6.25 per cent. for the period 1920–1925.

A method is outlined by which a measure can be obtained of the contribution of any particular sire to the graph of the average percentage of inbreeding for the breed. It also measures the degree of concentration of his blood in animals inbred to him and the rate at which his blood is diffused through the breed.

The portable gold-leaf electrometer designed by C. T. R. Wilson gives a means of measuring the charge upon and current through a conductor exposed to the earth's field and maintained at zero potential. Further, Wilson has shown (1) that the dissipation factor (the ratio of current per minute to the corresponding charge) is approximately the same for a surface of turf and the metal test-plate; and (2) how to deduce the corresponding charge and current per square centimetre on the neighbouring ground-level. Wilson's measurements were made chiefly in a country atmosphere (near Peebles), and we have made observations with a similar instrument in town air, in and near Edinburgh, to find out whether the dissipation factor is notably affected by the purity of the atmosphere as regards smoke, etc.

Professor Tait has shown that accurate determinations of surface-tensions would be obtained if we could measure the rate of propagation of ripples set up by the vibration of a tuning-fork of known pitch.

Professor Becker's paper “On the Spectrum of Nova Persei and the Structure of its Bands,” recently published in the Transactions of this Society, contains an interesting confirmation of some results already pointed out by Messrs Campbell and Wright of the Lick Observatory, which seem to be of considerable importance for the theory of temporary stars. By most careful micrometric measurements of the positions of the bright and dark bands in the photographic spectrum of Nova Persei, Professor Becker arrives at the conclusion that all the bands are similar in type, and that the distances of corresponding maxima and minima from the centres of the bands are proportional to the wave-lengths. The results derived from the Lick photographs point to exactly the same conclusion. It appears, therefore, from these two carefully and independently executed series of observations, that the chemical nature of the elements, whose light-vibrations gave rise to the selective radiations and absorptions noticed in Nova Persei, had no influence on the appearance of the bands. According to the Lick observers, there is no evidence that the structure and character of these bands were affected by other considerations than that of wave-length.

The form described in the present paper was found amongst the contents of the stomach of a Weddell seal (Leptonychotes weddelli), taken by the Scottish National Antarctic Expedition in Scotia Bay, South Orkneys. In all about sixty specimens were collected. They occurred unattached amongst material in a semi-fluid condition, so that it is not possible to state definitely whether the seal or some animal upon which it fed is the normal host of this parasite. This question, however, in the case of the Echinorhynchi, appears from the work of de Marval to be one of minor importance. He has shown that in the Echinorhynchi of birds there is a most marked absence of specialisation of hosts, and quotes from other authors various instances of the same parasite occurring in different vertebrate classes. The worms in the present case showed no indication of having been affected by the digestive juices of the seal. As will appear from the description given below, their structural peculiarities suggest relationships with certain forms known to inhabit aquatic birds.

The paper derives some new Bessel function inequalities These are used to determine conditions for the absence of roots of the transcendental equation F(z)Zv(z)+Zv′(z) = 0, where Zv(z) is a Bessel function, in certain regions of the v and z-plane

Order-of-magnitude results are extended to the case of general second-order term, with coefficient not necessarily of fixed sign, with general positive weight-function. The bounds are used to establish the expression for the Titchmarsh–Weyl function m(λ) as a Nevanlinna function in terms of the spectral function.

In the course of a recent investigation on the fertilised egg of the mouse (Gresson, 1940 a, b) it was necessary to keep a record of the time at which pairing took place. This revealed that copulation, especially in the winter, was very infrequent during the hours of daylight, and usually took place at about 12 midnight or between midnight and early morning. Similar observations were made by Lewis and Wright (1935), who found that in the autumn and winter mice seldom copulated during the day, and that while the majority of pairings took place between 10 p.m. and 1 a.m., several of their mice paired between 1 a.m. and 7 a.m., whereas in the spring pairing occurred more readily during all hours of the day.

It is shown that every weakly l1-stable linear and bounded operator (which represents a linear discrete-time system) on a Hilbert space is power stable. It solves (at least partially) a discrete-time version of a problem posed by A. J. Pritchard and J. Zabczyk for strongly continuous semigroups of bounded linear operators.

We continue with the work of an earlier paper concerning the use of partial differential equations to prove the uniform convergence of the eigenfunction expansion associated with a left definite two-parameter system of ordinary differential equations of the second order.

Let N(u, g, h) denote the number of subgroups of index u, genus g and parabolic class number h in the classical modular group. In 1974, Petersson proved that, for g = 0 and h = 1, 2, N(u, g, h) grows exponentially with u. We obtain a formula valid for all g and h. From this we derive an asymptotic estimate, showing that Petersson's result gives the correct type of growth. The proofs use coset diagrams.

We obtain estimates for the exponential growth of the solutions to u″(t) = (A + ζ2I)u(t) in terms of the exponential growth of the solutions to u″(t) = Au(t), where ζ is an arbitrary complex number. Estimates in exponentially weighted L2 norms are also considered in Hilbert space.

This paper deals with conditions which guarantee that a meromorphic function on the plane cannot satisfy any algebraic differential equation having coefficients in a given field of meromorphic functions. Some of the conditions are of growth type, while others depend on a representation for the function.