Until a few years ago it was the general opinion among metallurgists that the presence of manganese in steel exceeding the proportion of 1 per cent, is prejudicial to the value of the steel, inasmuch as a higher percentage of manganese has the effect of lowering markedly its tensile strength and toughness. But in 1884, Messrs Hadfield & Company, of the Hecla Steel Works, Sheffield, exhibited, at a meeting of the Institute of Mechanical Engineers, a number of samples of steel containing upwards of 10 to 15 per cent, of manganese, and submitted the results of experiments, which showed that the samples were, in point of tensile strength and hardness, in no way inferior to steel. Again, in 1888, Mr R. A. Hadfield read to the Institute a paper on the subject, giving the details of a large number of tests, which brought to light some interesting mechanical properties of alloys of manganese and iron. Since its introduction, these alloys (and particularly that containing 10 to 15 per cent, of manganese, known as “manganese-steel”) have been studied by several physicists, and further peculiarities have been found. It appeared desirable that the thermal conductivity of so peculiar a substance should be investigated. The present paper is an account of experiments made in the Physical Laboratory, Edinburgh University, with a view to the determination of its thermal conductivity. In the reduction of such experiments a knowledge of the specific heat is necessary, hence there is also given an account of experiments whereby the specific heat was determined.

The preliminary notices published by me in 1870 and 1872, on the action and chemistry of Strophanthus, indicated that it was likely to prove of value as a therapeutic agent; and so early as the year 1874, I had applied the substance in a few cases to the treatment of disease. Before sufficient data, however, had been obtained to justify any conclusions regarding its value as a therapeutic agent, the observations were interrupted by my leaving Edinburgh to occupy a public office in England, in connection with which it was impossible to conduct observations on the treatment of disease. In 1879, opportunities were again afforded to resume the interrupted observations, and results confirmatory of the anticipations which had been raised by the earlier physiological observations were gradually collected. The publication of a few of these results at the Cardiff meeting of the British Medical Association in 1885 has led to Strophanthus gaining a wide recognition as an important therapeutic agent, and to the production of numerous papers dealing with its botany, chemistry, pharmacology and therapeutics, not only in this country, but also in the continent of Europe and in America.

I have explained at some length, in my “Reply to Prof. Boltzmann,” the circumstances under which the present inquiry originated and has been pursued. Of these I need now only mention two:—first, the very limited time which I can spare for such work; second, the very meagre acquaintance I possessed of what had been already done with regard to the subject. My object has been to give an easily intelligible investigation of the Foundations of the Kinetic Theory; and I have, in consequence, abstained from reading the details of any investigation (be its author who he may) which seemed to me to be unnecessarily complex. Such a course has, inevitably, certain disadvantages, but its manifest advantages far outweigh them.

The purpose of the present paper is to ascertain how far it is possible to find exact solutions or values of x, y, &c., in equations between variables, so that the forms of plane curves and contour-lines of surfaces may be exactly determined. No approximate methods have been admitted, and only those methods have been used which are applicable to algebraic equations of every degree and any number of variables. In the examples given I have generally selected equations of even degree and even powers of the variables. But every such solution evidently includes the solution of the non-central equation of half the degree having corresponding terms and equal coefficients. The methods of solution employed are founded on the following introductory theorem or principle, which may be described as that of Homogeneous or Linear Variation of the quantities.