The first part of the paper is devoted to the computation of the strengths of the parts of a structure destined to resist given strains, taking into account, along with those strains, the unknown weights of the parts. The results obtained by this process necessarily give the best possible arrangement of the strengths, since, if any one part were made weaker, the whole structure would be weakened; or, if a part were made stronger, the unnecessary weight thus thrown upon the other parts would also go to weaken the fabric. It is believed that this investigation has now been given for the first time.

It was pointed out that this method enables us to determine the utmost limit of magnitude of a structure having a given general configuration.

This paper is a sequel to several communications which have already appeared in the Proceedings and Transactions of the Royal Society of Edinburgh. It commences with an investigation of the circumstances under which a portion of an incompressible frictionless liquid, supposed to extend through all space, or through space wholly or partially bounded by a rigid solid, can be projected so as to continue to move through the surrounding liquid without change of shape; and goes on to investigate vibrations executed by a portion of liquid so projected, and slightly disturbed from the condition that gives uniformity. The greatest and least quantities of energy which a finite liquid mass of any given initial shape and any given initial motion can possess, after any variations of its bounding surface ending in the initial shape, are next investigated; and the theory of the dissipation of energy in a finite or infinite frictionless liquid is deduced.

Le Sage, born at Geneva in 1724, devoted the last sixty-three years of a life of eighty to the investigation of a mechanical theory of gravitation. The probable existence of a gravific mechanism is admitted and the importance of the object to which Le Sage devoted his life pointed out, by Newton and Rumford in the following statements:—

“It is inconceivable that inanimate brute matter should, without the mediation of something else, which is not material, operate upon, and affect other matter without mutual contact; as it must do, if gravitation, in the sense of Epicurus, be essential and “inherent in it.

While engaged in some quaternion researches with reference to Spherical Harmonics, which I hope soon to lay before the Society, I was led to imagine that some of my results might produce a simplification of the ordinary modes of treating the subject. The following is the result of the attempt. It seems to make the calculus somewhat more intelligible to the beginner than the methods employed by O'Brien and Murphy, whose works on the subject are usually read in this country. As I am not writing a treatise, but merely sketching a method, I shall run over the principal elementary propositions only.

For some time back I have been endeavouring to prove, by experiment, through great ranges of temperature, the result announced by me in December last, viz., that the electro-motive force of a thermo-electric circuit is in general, unless the temperature be very high, a parabolic function of the absolute temperature of either junction, that of the other being maintained constant.

The following was the arrangement adopted in these experiments :—A large magnet was put into a copper pot containing oil, which was heated up by a brass Bunsen, and its temperature determined by a mercurial thermometer immersed in it. The magnet was set magnetically east and west, and placed so as to act with equal force on the poles of a small magnet, whose centre was in the prolongation of its axis. This small magnet was cemented to the back of a small concave mirror, suspended by a single silk fibre, and placed in a glass case to guard it against currents of air. The deflections of the small magnet were measured exactly as in the reflecting galvanometer, and since from the nature of the arrangement, the absolute magnetism in the large magnet is directly as the tangent of the angle of deflection of the small one, its amount for any temperature was immediately measured by the reading on the scale.

While suffering some of the annoyances seemingly inseparable from re-vaccination at too advanced an age, I was led to the curious observation presently to be described. I was unable to sleep, except in “short and far between” dozes, from which I woke with a sudden start, my eyelids opening fully. I found by trial that this state of things became somewhat less intolerable when I lay on my back, with my head considerably elevated. In this position I directly faced a gas jet, burning not very brightly, placed close to a whitish wall, and surrounded by a ground glass shade, through which the flame could be prominently perceived. The portions of the wall surrounding the burner were moderately illuminated, and hyperbolic portions above and below somewhat more strongly. I observed, on waking, that the gas flame seemed for a second or two to be surrounded by a dark crimson ground, though itself apparently unchanged in colour. Gradually, after the lapse of, at the very utmost, a couple of seconds, everything resumed its normal appearance. As this phenomenon appeared not only to be worthy of observation in itself, but to furnish me with something definite to reflect upon, which is far the best alleviation of annoyances similar to those from which I was suffering, I determined to watch it, transitory as it was, feeling assured that I should have many opportunities of observing it.

By combining, as above, Hamilton's linear and vector-function with his celebrated vector square-root of the negative of Laplace's operator, an operator of great use in physical applications of mathematics is obtained. With the notation employed in the author's paper “On Green's and other Allied Theorems,” Trans. R.S.E. 1870, § 17, it is shown to be generally expressible in the form of

where α,β,γ, are any three unit vectors (not necessarily rectangular), and α1,β1,γ1 any three vectors whatever.

Mr Sang's papers in recent parts of the Transactions of the Society have reminded me of some geometrical constructions which are to a certain extent indicated in Tait and Steele's Dynamics of a Particle (1856). Some of these were suggested to me by a beautiful construction given (I believe by Clerk-Maxwell) in the Cambridge and Dublin Math. Journal, Feb. 1854, the others by a very simple process which occurred to me for the treatment of oscillations in cycloidal arcs. The former enables us easily to divide the arc of oscillation of a pendulum, or the whole circumference if the motion be continuous, into two, four, eight, &c., parts, which are described in equal times; also to solve by simple geometrical constructions problems such as the following: —Given any three points in a circle, find how it must be placed that a heavy particle, starting from rest at one of them, may take twice as long to pass from the second to the third as it takes to pass from the first to the second.

The principles set forth in this paper, though now (with the exception of the first theorem) published for the first time, were communicated to the French Academy of Sciences fifteen years ago, in a memoir entitled “de I Equilibre intérieur d'un Corps solide, élastiqne, et homogène,” and marked with the motto, “Obvia conspicimus, nubem pellente Mathesi,” the receipt of which is acknowledged in the Comptes Rendus of the 6th April 1857 (vol. xliv. p. 706.)

The author quotes a theorem discovered by him, and previously published in the Philosophical Magazine for December 1855, called “the Principle of Isorrhopic Axes,” viz., “Every self-balanced system of forces applied to a connected system of points, is capable of resolution into three rectangular systems of parallel self-balanced forces applied to the same points.”

The class of curves resulting from the formula

are of considerable interest as occurring in various mechanical inquiries. When a straight wire, whose effective breadth and thickness are as two to one, is fixed at one end and made to vibrate, its free end describes a curve of which the general equation is

in which k is constant for the particular variety of curve. When the curve becomes a parabola, and when k = 0, it takes the form above mentioned; these phases were exhibited by me in 1832. Again, when a system of toothed wheels is deduced from a straight rack, having a curve of sines for its outline, the points of contact describe a curve of this class, as is shown in my treatise on the teeth of wheels.

During the deep sea dredging expedition of H.M. ships ‘Lightning’ and ‘Porcupine,’ in the summers of 1868–69 and 1870, two or three nearly perfect specimens, and a number of fragments were procured of three species of regular echinideans, which were referred by the author to a new family, the Echinothuridæ, intermediate in their more essential characters between the Cidaridæ and the Diadematidæ.

In these urchins the test is circular and greatly depressed. The plates of the perisom are long and strap-shaped, and the interambulacral plates overlap one another regularly from the apical towards the oral poll, while the ambulacral plates overlap in a similar way in the opposite direction.

It is well known that many taxes do not fall ultimately on the person from whom they are in the first instance levied. The merchant advances the duties imposed on goods, but the tax ultimately falls on the consumer. The problem of discovering the ultimate or true incidence of each tax is one of great importance, and of considerable complexity. The following paper contains an investigation of the methods by which this incidence may in some cases be experimentally determined, and of the principles regulating the incidence in all cases, these principles being stated in a mathematical form.

In a communication made to this Society on the 6th February, 1871, I noted the capture of a sperm-whale at Oban in May, 1829, and I collected from various sources records of the stranding of seven additional specimens on the Scottish coasts.

Since that communication was published, a large sperm-whale has come ashore on the west coast of the Isle of Skye, some particulars concerning which I propose to relate in this communication.

Tourists in Skye, during the past autumn, who visited Loch Corruisk by boat from Torrin, as they sailed up Loch Scavaig, became conscious, by another sense than that of sight, that a large animal in a state of putrefaction was in their immediate vicinity.

The following observations on Cystine are a continuation of those formerly communicated to the Society by Dr Arthur Gamgee and myself, during the course of the Session 1869–70, and reprinted with addition in the ‘Journal of Anatomy and Physiology’ for that year; and although really little of a novel nature to present to the Society, still it is necessary to show some additional facts have been observed tending towards the synthesis of this interesting substance.

The observer of the current literature of Midwifery finds nothing more characteristic of it than the number of papers on the mechanism of natural parturition. These papers indicate for the most part an enlightened zeal, for they are engaged with a most important branch of this mechanism, namely, the mode of action of the force of labour upon the fœtus and upon the passages, and the explanation thereby obtained of the changes which take place in these as natural labour advances.