The first two parts of this paper—Physical Geography and Salinity—were communicated to the Society on May 18th, 1891, and published in the Transactions, Vol. XXXVI, Part III., No. 23, pp. 641–729.

Various circumstances have prevented me from sooner presenting the concluding part of the discussion. I postponed publication again and again, in the hope that it might be possible to discuss the results more thoroughly, and deduce from them more clearly than I have been able to do the laws which regulate the heat-transactions of sea-water of varying salinity, contained in basins of differing degrees of isolation from the circulation of the ocean. At length the conclusion has been arrived at that the observations are not sufficiently uniform, regular, and close to warrant the expenditure of the time devoted to their discussion. Many months of work have been occupied in proving that some special manner of classifying and treating the data led to no definite result. Thus, it is unnecessary to describe several series of voluminous calculations, or to bring forward a great number of maps and sections on which the distribution of temperature was plotted in different ways. It is difficult to establish theoretical conclusions of a general and far-reaching kind from my work, and I have not attempted to compare it with the many memoirs published in continental journals, on the temperature of lakes, fjords, and enclosed seas.

Systems of ordinary linear differential equations are of great importance, both from a practical and from a theoretical point of view. They figure largely in dynamical problems; and Jacobi has shown that the general problem of determining the order of any system of ordinary differential equations whatever can be reduced to the problem of determining the order of a linear system with constant coefficients. Nevertheless, the present state of the theory of such a system still leaves something to be desired. It is true that a logical and systematic process for the solution was given by Cauchy. This consists in first replacing the system by another in which only first differential coefficients occur, by introducing as auxiliary variables the successive differential coefficients of the various dependent variables up to the highest but one, and then reducing this system to the “normal form” by calculating the differential coefficients as linear functions of the dependent variables. It happens, however, when we attempt to do this, that we are led to a system consisting partly of differential equations of the form

where fr, denotes a linear function of x1, …., xs, partly of a number of non-differential equations connecting the remainder of the variables with x1, …., xs. The order of the system—that is to say, the number of independent arbitrary constants required for its complete solution—is the number of differential equations in the normal form; but no rule is readily deducible from the method for determining beforehand how many of the equations in the normal system will be differential equations, so that we cannot predict the order of the system without actually going through the labour of reduction. Moreover, the normal form is in practice often not the most convenient for the purposes of solution.

The following essay, except for one or two slight corrections and additions which I have since interpolated, was read before the Eoyal Society of Edinburgh in June 1894. Very shortly thereafter, M. Jean Svoronos published in the Bulletin de Correspondance Hellénique (Janv.–Juillet, 1894) a learned paper “Sur la signification des types monétaires des anciens,” in which he demonstrated with an elaborate wealth of illustration the astronomic significance of many coin-types, the precise point that the greater part of this essay of mine was written to prove. Beginning with coins on which a beast- or bird-emblem is figured together with the symbol of a star, and passing on to others where the star-symbol is omitted, M. Svoronos shows clearly that in a very varied series of coin-types, the Lion, the Bull, the Eagle, the Horse, and so forth, represent not merely these creatures themselves, but their stellar namesakes: in short, that, in more or less obscure and occult shapes, astronomic emblems are imprinted on a vast range of ancient coins, just as in open and acknowledged forms they are visible, for instance, on the coins of Antoninus Pius. So clearly is all this put forward by M. Svoronos that my paper might well have remained unpublished were it not that I think I take the case somewhat further than he does. For, whereas M. Svoronos is content to demonstrate the symbols of individual constellations, I have attempted also in certain cases to show that the associated emblems correspond to the positions relative to one another of the heavenly bodies, in some cases to the configuration of the sky at critical periods of the year or at the festival seasons of the cities to which the coins belong. In some other respects, also, I have attempted to carry to a further issue the general considerations suggested by the astronomic hypothesis.

The group of questions which are connected with the Glacial Age seem to me to be among the most interesting and the most difficult in the whole science of geology. They include the question of Time—since, perhaps, the only hope we have of even reaching any unit of time in geological changes lies in the phenomena of the Glacial Age, The question of the comparative slowness or suddenness of great physical changes is not less directly involved. Bound up with this again is the question of the sudden or slow destruction of the extinct forms of life, and the introduction of new forms to replace them. The connection between Cosmical and Terrestrial causes of change comes directly into our view, and then the nature and operation of the terrestrial forces which were brought into play. I am very sceptical as to many of the solutions which have been proposed for most of these questions, and still more for the theories which profess to answer them as a whole. There is nothing to be done but to accumulate evidence in detail—to observe facts well, that is, completely—and avoid looking only at such of them as tell in favour of some preconceived hypothesis.

For many years the Fossil Flora of the Yorkshire Coal Field has been engaging my attention, and among the species occurring in that district are many of considerable interest. This Coal Field supplied Artis with the specimens which he figured and described in his Antediluvian Phytology.

In 1888, at the Annual Meeting of the Yorkshire Naturalists' Union, held at Malton, a committee was formed for the investigation of the Fossil Flora of Yorkshire, and since that date four Reports have been prepared and published based upon specimens submitted to me for examination by private collectors, and from collections contained in public museums. These Reports only contain lists of the species found, and the localities and horizons from which the specimens were derived, with any occasional short notes that might have been thought necessary. All detailed descriptions or critical remarks were deferred, and the present paper is the first of what I hope may be several, dealing more in detail with the Fossil Flora of the Yorkshire Coal Field.

Clerk Maxwell, in his Electricity and Magnetism, vol. i. § 304, makes the following remark about the rotatory coefficient:—“It should be found, if anywhere, in magnets which have a polarisation in one direction, probably due to a rotational phenomenon in the substance.”

The current which should arise from such a coefficient was first observed by Hall. He passed a current through a strip of metal; he then found two points on opposite sides of the strip, which, while the current was flowing, were at the same potential, and which therefore indicated no current when joined to a galvanometer. The plate was next brought into a uniform magnetic field, and when everything was steady the two points previously at the same potential were no longer so, and a current flowed through the galvanometer. This effect is observable in all conductors.

Kundt has shown that the transverse effect in iron, cobalt, and nickel is proportional to the magnetisation. Such an effect, where the magnetisation appears in the first power, we shall call a Hall effect. In applying the same method to bismuth, he found that no transverse effect was given by the thin plates of the electrolitically deposited metal used by him. That this absence of transverse effect is not characteristic of all bismuth so prepared has been shown experimentally. The question is, What relation, if any, exists between it and the magnetisation? To settle this it is necessary to compare the transverse effect in any given plate with some other effect in the same plate whose relation to the magnetisation is known. Such an effect is the variation of resistance. Goldhammer has shown that this latter is proportional to the square of the magnetisation.