In the present communication I have applied the results of my first paper to the question of the transference of momentum, of energy, and of matter, in a gas or gaseous mixture; still, however, on the hypothesis of hard spherical particles, exerting no mutual forces except those of impact. The conclusions of §§ 23, 24 of that paper form the indispensable preliminary to the majority of the following investigations. For, except in extreme cases, in which the causes tending to disturb the “special” state are at least nearly as rapid and persistent in their action as is the process of recovery, we are entitled to assume, from the result of § 24, that in every part of a gas or gaseous mixture a local special state is maintained.

The annexed tables are intended to facilitate the computation of the corrections for refraction which have to be applied to differential measures, such as are made with the micrometer or heliometer.

Differential measures are of two kinds :—(1) Direct measures of differences of right ascension and declination; and (2) measures of position angle and distance. In either case the observer only seeks to determine the relative positions of the objects under observation; and the correction for refraction consists in the applying to each reading a quantity representing the difference of the separate effects of refraction on the apparent places of the two stars, whose relative positions are to be determined.

A glance at the expression

is sufficient to verify the fact that it is symmetric with respect to a, b, c, d, and also with respect to α, β, γ, δ. It is likewise, although not quite so evidently, an alternating function with respect to the interchange

that is to say, if a and α be interchanged, and at the same time b and β, c and γ, d and δ, the function is not altered in magnitude, but merely changes sign. With a little trouble, indeed, the expression can be transformed into

or say

The expansion of ϕ(x) in terms of Go(x), G1(x), G2(x), &c., connected by a given law. being of great importance in mathematico-physical investigation, every method of effecting this expansion must have some interest for scientists.

I therefore proceed to propose what I think to be a new method, in the hope that it may prove to be useful.

Many special expansions of this nature have been effected by Fourier, Legendre, and others.

In the course of some remarks on the scheme proposed for the Forth Bridge, which remarks are published in the eleventh volume of the Transactions of the Royal Scottish Society of Arts, I was led to enunciate, among other theorems, one of a somewhat unexpected character, to the effect that any symmetric structure built on a rectangular basis, having no redundant parts, and depending on longitudinal strain alone, is necessarily unstable. This theorem was established by arguments restricted to the single matter under consideration; it is one of an extensive class, and I now propose to discuss the subject from a general abstract point of view.

My attention for the last few years having been specially directed to the vertical distribution of the Carboniferous Fossil Flora, it is my intention to publish a series of papers dealing with this subject.

While carrying on these investigations it has been necessary, in addition to visiting public and private collections, to visit several of the coal fields for the purpose of collecting specimens, as in almost no case have the smaller and less attractive species been secured, and, as a rule, only what strikes the ordinary collector as being “a fine specimen” is preserved, to the exclusion of many less striking but often more valuable examples. Hence our public collections, and, with few exceptions, also our private collections, give a very imperfect idea of the richness of the flora of the Carboniferous Formation of Britain.

The sample of the remarkable deposit of Diatomite, upon which the subjoined observations have been made, was forwarded to me some time ago by William Morrison, Esq., of the Academy, Dingwall, through my friend Mr John Gunn. With the exception of a brief notice in the newspapers soon after the discovery of the deposit by Mr Morrison a few months ago, and a paper entitled “On some New Localities for the Mineral Diatomite, with Notes on the Chemical Composition of the Specimens exhibited,” in the Mineralogical Magazine and Journal of the Mineralogical Society, vol. vii. No. 32, pp. 30–34, July 1886, by W. Ivison Macadam, F.C.S., F.I.C., Professor of Chemistry, New Veterinary College, Edinburgh, no statistics have been published in connection with the deposit, nor has any account hitherto been given of the Diatomaceœ which play so prominent a part in its composition.

The structure of the eye in Isopods has been less studied than in any other group of Arthropods; the only modern descriptions of the minute anatomy of the eye in these crustaceans known to me are by Grenacher of Porcellio, by Bullar of Cymothoa, by Bellonci of Sphæroma, by myself of the genus Serolis. I have lately had the opportunity, while engaged upon my Report upon the “Challenger” Isopoda, of investigating the eye in several species of Æga and allied genera. The structure of the eye in these Isopods differs very materially from the descriptions given by Grenacher and Bullar, but agrees very closely with the structure of the eye in Serolis.

The question is the following:—We consider the solid angle formed by three planes at right angles to each other, and into the space of this single octant we introduce a given ellipsoid, and cause its surface to be tangent to each of the three sides of the solid angle. The position of the points of tangence will of course be variable in each plane according to the orientation given to the axes of the ellipsoid, but it is evident that on each of the planes the positions of the point of contact will be unable to outpass certain limits so long as the ellipsoid fulfils the condition of remaining tangent simultaneously to the three planes: these limiting positions of the point of contact in one, as for example, of the planes, will form a certain curve, and the proposed question will be: the determination of that curve, the limiting curve as we shall call it in the Sequel.

At the suggestion of Prof. Tait, an attempt has been made in this paper to apply the method used by him in § 21 of his paper on “The Foundations of the Kinetic Theory of Gases” to a case of the question of the distribution of energy in a system of non-homogeneous impinging spheres.

The problem may be stated as follows:—Given a very great number of smooth elastic spheres, equal and like in all respects, whose centres of figure and centres of inertia do not coincide, and the sum of whose volumes is but a small fraction of the space in which they move, it is required to find the ultimate distribution of energy among the various degrees of freedom when by collisions the system has attained a “special state.”

Since its discovery by Dumas and Péligot in 1834, methyl-alcohol has been the subject of a great many researches, and as a result we have long had a perfectly certain knowledge of its atomic composition, and a very accurate knowledge of a great many of its reactions. Yet the physical properties of the substance CH4O have not yet been determined with a satisfactory degree of precision. At this we need not wonder. For the study of the transmutations of a species a very impure specimen may suffice, and a series of such studies may leave no doubt about the correct atomic formula of the species in question, and consequently also, if it is a volatile substance, about its perfect gas density. But no other physical properties can be determined otherwise than by direct experiments on a pure specimen. And pure methyl-alcohol is very difficult to obtain. In whatever reasonable sense we may take the word “pure,” as attached to the name of a chemical preparation, “pure” methyl-alcohol must be admitted to have been little more than a chemical fiction until Wöhler in 1852 discovered his well-known (oxalate) process for its extraction from wood-spirit.

Shortly after I read to the Society my paper on “Thermal and Electric Conductivity” (Trans. R. S. E., 1878), in which I stated that the results were “by no means final, even so far as my own work is concerned,” I was requested by Sir Wyville Thomson to undertake the examination of the “Pressure Errors of the ‘Challenger’ Thermometers.” This investigation led to another on the “Compression of Sea-Water,” and allied subjects, which is not yet finished. Meanwhile, though I had prepared everything for my promised repetition of the experiments on Thermal Conductivity, the bars formerly used having been nickelised, &c, I found that it would be impossible for me to carry out the investigation. I therefore asked Mr Mitchell, who, as Neil-Arnott Scholar, had already done good and careful work on Thermal Conductivity in my Laboratory, to repeat the experiments under the altered conditions. I put at his disposal all the apparatus which was employed in the former research. The Government Grant Committee allowed a sum for the payment of a computer to reduce the results, and the observations were at once commenced. The results are now laid before the Society, and are probably as good as the method and the thermometers employed can furnish.

The analytical methods referred to in our heading are infected with numerous sources of error, amongst which until lately the uncertainty of our knowledge of the combining constant “Pt” played a prominent part. This uncertainty, it is true, has been removed to a great extent by Seubert's investigation “Ueber das Atomgewicht des Platins,” Liebig's Annalen (for 1881), vol. ccvii. p. 1 et seqq. The value 194·8 for Pt, which he ultimately adopted, falls in well with his analyses of the chloroplatinates of potassium and ammonium, and there can be no doubt that his chloroplatinates were close approximations to the ideal substances. Hence his atomic weight 194·8 must be admitted to be nearer the truth than the value 198 of Andrews, which, until lately, was so generally employed by chemists in the calculation of their analyses. But it does not follow that in, for instance, the determination of chloride of potassium “as metallic platinum,” the factor 2KCl : Pt = 0·7657, calculated from Seubert's Pt, gives a more correct result than even the factor 0·75252, which follows from K = 39 : Cl = 35·5, and Pt = 198.