A differential equation may be regarded from two points of view, one purely analytical, the other geometrical. From the analytical point of view, a differential equation of the first order is merely a functional relation between x, y, and p (where p = dy/dx), and the problem of solving the equation is to find a function of x, say f(x), such that if f (x) and df(x)/dx are substituted for y and p in the equation, the result is an identity in x. In the geometrical method, on the other hand, x and y are treated as the co-ordinates of a point in a plane and p as a direction. The differential equation then attaches to every point in the plane a certain direction, which may be conveniently represented by an arrow drawn through the point. The problem of integration then resolves itself into finding a family of curves, such that, at every point (x′, y′), the direction of the curve at that point is the direction obtained by substituting x′ and y′ in the differential equation and solving for p. These curves are called the integral curves of the equation. This method owes its development chiefly to Lie.

The general scalar function of a vector ρ may evidently be written in the form

where α1 … αn are constant vectors, each term being homogeneous in ρ. If the factors of each term of any one of the brackets be permuted in every possible way, i.e., in |r ways (where r is the degree of the set of terms chosen), it can be written in the form

where φρr−1 is a vector function of ρ of the (r − l)th degree, the index (r − 1) serving merely to indicate the degree of the function. This notation will be found to lead to no confusion, and to have great advantages. Since the variable ρ occurs in φ along with each vector α, and also these vectors have been arranged in every possible way, it follows that, if in each term of φ we replace s of the ρ's by any other vector σ in such a way that the order of occurrence of the ρ's and the σ's is the same in each term, then the final result is independent of that order; that is to say,

The equation to the pair of tangents from the point (x′, y′) to the conic φ(x, y) = 0 is usually obtained in the form

where T = 0 is the equation to the chord of contact. This equation admits of reduction; and we propose to obtain the reduced form independently, and to supply its geometrical interpretation.

During the discussions that have arisen as to the internal heat of the earth, the writer has never seen any reason given for supposing that there was a time when the earth was a highly heated fluid mass, and he believes that view to have originated by analogy from the case of the sun; and no other cause of the heat seems to be generally assumed than the collisions of the parts that came together to form the earth's mass. He has expected to find some one maintaining that gradual gravitational compression of the mass was the main source of the earth's internal heat, but till recently he has never tried to find out if it could possibly be a sufficient cause. It has always seemed to him that the formation of the earth's mass must have been accomplished under circumstances so different from the case of the sun that an analogy could scarcely be drawn between the two cases; indeed, it is obvious that the amount of heat produced by the formation of planets from nebulae will depend principally on their masses, and will be in a, higher ratio than that of the masses.

In August 1901 Mr Thomas Anderson, merchant, Hillswick, to whom and to other members of whose family I have on several occasions been indebted for specimens to illustrate the Zoology of Shetland, wrote to tell me that a dead whale had been found floating near Hillswick and had been towed into Roeness Voe. It was within three miles of land when seen by the fishermen and was claimed by the Crown. From the appearance of the animal Mr Anderson thought that it was a sperm whale, and he sent me the following description, from which it was clear that he had correctly recognised it.

During the winter of 1902–3, I conducted observations upon the scales and their condition, in several of the gadoid fishes. Investigation was commenced in October 1902; but it was not until the beginning of March 1903 that the first appearance of the young scale took place. In stained specimens, it can be recognised as a deeply staining “nucleus,” lying beneath the old scale, just under its centre. Such an appearance was found in cod, haddock and whiting of all ages from one to three or four years;, and, in all, the young scale is clearly recognisable, underlying the old. As soon as these fishes have spawned, they appear to shed their scales, the epidermis first peeling off.

F([α] [β] [γ] denotes the convergent infinite product

which may be expanded in an infinite series

The theories which give the best account of the facts of colour vision and colour blindness are the Young-Helmholtz theory and Hering's theory. Both are trichromatic theories, and, apart from physiological or anatomical questions, both can, by proper choice of fundamentals, be made to give a good account of the main facts. The facts of one-eyed colour blindness show that, on the Young-Helmholtz theory, colour blindness must be regarded as due to fusion of at least two fundamental sensations, but the curves of one sensation determined by observations on different eyes differ considerably among themselves, and this indicates that a broader basis for the theory may be desirable. This may be sought for in a tetrachromatic theory, but any such theory must explain the possibility of trichromatic representation of all colours. A theory proposed assumes two pairs of complementary stimulations, say R,G1 and G2,V.

In this method the liquid of which the latent heat is to be determined is contained in a glass vessel which is suspended by a fine wire from the hook of a balance, and the amount of electrical energy is measured which is required for the evaporation of an observed mass. The electrical energy is supplied by a large current passing through a small resistance in the liquid itself. In order to reduce heat losses as much as possible, the vessel carrying the liquid and the resistance is completely surrounded by a doublewalled shield, which is filled by the saturated vapour of the liquid itself. The estimation of the electrical energy supplied necessitates tne measurement of the current, C (amperes), the resistance R (ohms), of the heating coil, and the time in seconds, T, during which the current flows. It is expressed by C2RT joules, and if M be the mass evaporated in gms., the latent heat expressed in joules per gm. becomes C2RT/M.

In the time of Pallas and Pennant, as in the days of Oppian and Pliny, it was commonly believed true wild horses were to be met with not only in Central Asia, but also in Europe and Africa. But ere the middle of the nineteenth century was reached, naturalists were beginning to question the existence of genuine wild horses; and somewhat later, the conclusion was arrived at that the horse had long “ceased to exist in a state of nature.”

The materials for this paper upon the Influence of Artificial Propagation on the Salmon of the American Rivers have been gathered from an examination of the salmon fishery statistics of Canada and the United States of America, where salmon culture has been extensively pursued.

While, as a result of recent investigation, a good deal is now known of the life history of salmon in fresh water, we possess little knowledge of their mode of existence in the sea.

In Spectro-Photometry one of the chief difficulties in making accurate measurements lies in the fact that the eye has to judge of the relative intensities of two spectral bands, which, instead of having their adjacent edges in contact, have them separated by a greater or less dark space. Accordingly, to facilitate a piece of spectrophotometrical work in which I am engaged, a Hüfner's rhomb was tried, which is a special attachment designed to overcome this difficulty. On trial, however, I found that by this means the fault is not entirely removed, although the separating dark space is by this means reduced to a very narrow line. The only alternatives to the Hiifner's rhomb which I know are the devices of Brace, and of Lummer and Brodhun. The first of these depends on the employment of a particular kind of prism in the spectroscope. The other device, due to Lummer and Brodhun, places before the ordinary spectroscope prism a special photometer cube. Both these devices, however, suffer from the drawback that they are unsymmetrical as regards the two beams of light, whose relative intensities it is our object to measure.

The following result, which forms a part of a prolonged investigation into the relations existing among the quantities, magnetization, resistance, and temperature of the magnetic metals, seems sufficiently novel to merit a short note. The results for longitudinal magnetization were communicated to the Society on May 4, 1903; and a somewhat similar form of apparatus was set up in order to study the effect of high temperatures upon the connection between transverse magnetization and resistance in nickel wire. The smallness of the effect of resistance in nickel due to a transverse magnetic field, as compared with the effect of a longitudinal field, rendered all attempts to measure it with this form of apparatus quite nugatory. We were compelled to fall back upon the method employed by Kelvin, Goldhammer, and others, and use the air-gap of a fairly powerful electromagnet as the field.

The anatomy of the primates is responsible for a very considerable volume of contributions to morphological literature. But, quite naturally, those forms which have the closest affinity to man have been more extensively and more carefully observed than those occupying a lower position in the zoological scale. And, further, the most painstaking and accurate work has been done in connection with organs which are very specially modified in man, such, for example, as those of the nervous system. This is perfectly consistent with the desire to obtain any light, and from any source, which will facilitate the unravelling of the many problems of human structure. Moreover, the soft and flaccid viscera do not lend themselves to accurate observation so readily as do the more solid organs which retain their shape after their removal from the body. So it comes to pass that, up to quite recent times, the abdominal viscera, and more especially the hollow organs of the abdomen, such as the stomach and the intestines, have only received scant attention; or, if they have been more carefully examined, the descriptions as given prior to the introduction of modern methods cannot be held as absolutely satisfactory, because of the change of shape and disposition resulting from the removal of their natural support, to wit, the abdominal walls.

The demagnetisation of iron by decreasing reversals secures a molecular condition such that there is perfect symmetry to subsequent applications of either a positive or negative magnetising force (H). In a former communication it was shown that this symmetry is uni-directional. It is symmetry to the directional force (ab) used in the demagnetising process. During the early stages of induction iron is more permeable to a reapplication of a magnetising force in the same direction (+ or − ) as that used in the immediately preceding demagnetising process, than it is to a force (+ or − ) at right angles to that used in the immediately preceding demagnetising process. The hypothesis was advanced that on the completion of this process, a preponderance of the molecules would point towards an equatorial belt at right angles to ab. If H makes an angle other than zero with ab, the number of molecules lying in the most advantageous position, i.e., equatorially, is reduced, and the pull becomes less and less as the angle approximates to 90°.

(1) In his “Recherches sur les déterminants,” published in the Bulletin of the Royal Academy of Belgium in 1846, Catalan gives a proposition in regard to circulants of the type

which deserves some special attention, as there can be little doubt that it is seriously misleading. Unfortunately, too, it has actually already misled to a considerable extent, the proposition having been reproduced by quite a number of writers without any care being taken to verify it. Even the very latest text-book, that written by Professor Pascal of Pavia and translated into German by H. Leitzmann, gives importance to the proposition, and by omitting the so-called proof diminishes the chance of detection of error.

Attention has already been drawn in Part I. of this history to certain identities of Lagrange's which might possibly be viewed as contributions to the theory of determinants. Among these were the following published in 1773:—

The pineal body or epiphysis has provided a favourite subject of study for many workers at research, and much discussion has arisen with regard to both its ontogeny and its phylogeny. Up to within recent years this structure proved a sort of enigma, and its true significance was not really understood until the researches of De Graaf (15) and Baldwin Spencer (27) during the eighties. Since then an extensive amount of work has been published on the subject, so that it almost appeared quite unnecessary for one to again enter upon this field of research.