Minding has proved a remarkable theorem concerning a variable system of forces defined as follows:—the points of application of the different forces and their magnitudes are given, while the directions are such that a pencil of rays through any given point parallel to them moves as a rigid body.

In the “Journal für praktische Chemie,” [2] 6, 99, Pfankuch states that “when one adds chloroform to an alcoholic solution of sulphide of potassium a reaction takes place often with explosive violence, and by repeated extraction of the mass with absolute alcohol one obtains a compound which crystallises from alcohol in long prisms, and may be considered as a double salt of sulphide of potassium and sulphoform.”

I. If be a rational equation of the nth degree, Newton's rule for a superior limit to the number of its imaginary roots depends upon the changes of sign in the series of functions—called, by Sylvester, Quadratic Elements—

n + 1 in number.

It is a matter of some interest to know the relations in which the quadratic elements stand to the roots of the equation. The following method exhibits this relationship, and leads to others of a higher class.

In the autumn of 1877 I found on stones in a sea-water aquarium in Professor Huxley's laboratory at South Kensington, the curious little Alga which forms the subject of the present paper. I have ever since kept specimens growing, carried them with me to various places in Britain and on the Continent, and subjected them to frequent examination at all seasons of the year. Dr Bornet has kindly identified it for me as a species of Enteromorpha.

The following experiments, in continuation of our research on the discharge of electricity, were made in Professor Tait's Laboratory during the College Session 1878–79, and papers briefly describing the results were read before the Society on 6th January 1879 and 19th July 1880.

In the course of a series of experiments upon fusion point, as determined by the mercurial thermometer, I had occasion to investigate several of the properties of that instrument. In the hope that the results of that investigation may economise the labour of other experimenters, and aid in its accuracy, I have ventured to submit them to the consideration of the Society. The memoir comprises the following sections:—

I. Calibration and the calibration unit.

II. The exposure correction.

III. The movement of the zero with (α) time, and (β) temperature.

IV. Poggendorff's correction.

V. Comparison of the mercurial with the air thermometer.

VI. Effect of compression.

The following experiments were undertaken with a view to determine by actual observation the effect produced on solids by hydraulic pressure.

The instrument was constructed according to my directions by Mr Milne of Milton House, about two years ago, but it is only now that I have been able to devote myself to its application to the purposes for which it was designed. It consists of a hydraulic pump, which communicates with a steel receiver capable of holding instruments of considerable size, and also with a second receiver of peculiar form. This receiver consists essentially of a steel tube, terminated at each end by thick glass tubes fitted tightly. It is tapped at the centre with two holes, the one to establish connection with the pump, and the other to admit a pressure gauge or manometer. The steel tube may be of any length, being limited only by the extent of laboratory accommodation at disposal. The tube which I am using at present has a length of a little over six feet, and an internal diameter of about three-tenths of an inch. The solid to be experimented on must be in the form of a rod or wire, and must, at the ends at least, be sufficiently small to be able to enter the terminal glass tubes, which have a bore of 0·08″ and an external diameter of 0·42″. The length of the solid is such that when it rests in the steel tube its ends are visible in the glass terminations.

The alloys which we used in the following experiments were in the form of very thin hard-drawn wires,—the same as those whose thermo-electric properties we described in a paper published in the Transactions of this Society in 1877. They were prepared by Messrs. Johnson and Matthey of Hatton Garden, London. In giving their constitution we rely upon the authority of the manufacturers, as the quantities we had were too small to admit of our having them analysed.

Some time before the telephone was invented, I had occasion to consider very closely the problem of the opposition offered to the passage of the electric current by an electrolyte, and to seek for new methods of dealing with it. It was not difficult to see that the telephone afforded advantages in this kind of electrical measurement. As far as regards the measurement of what is usually called Electrolytic Polarisation, these advantages are perhaps even greater than they might at first sight appear. In the case of what is generally called Electrolytic Resistance, they are, however, less than they appear.

On the 26th of last month the full year appointed by Government Contract for the testing of the new Rock-Thermometers having expired, and they having approved themselves at all points, been accepted, and set fairly afloat on a new course of observation,—I hasten to announce the event to the Royal Society, Edinburgh, who have long had a lively interest both in these instruments and in the problems they have been employed upon.

In connection with the problem of Map-colouring, I incidentally gave (Proc. K.S.E. 1880, p. 502) a theorem which may be stated as follows : — If 2n points be joined by 3n lines, so that three lines, and three only, meet at each point, these lines can he divided (usually in many different ways) into three groups of n each, such that one of each group ends at each of the points.

Although, from the careful descriptions which have been given by several competent naturalists, we may now consider ourselves tolerably well acquainted with the structure of the dainty siliceous skeletons of this and several allied Hexactinellid Sponges, this is by no means the case with their soft tissues ; and the great cause of our imperfect knowledge of these interesting structures is that no observer has hitherto succeeded in procuring a really well-preserved Sponge of this group. It was with pleasure, therefore, that I accepted the offer of the Director of the Challenger Expedition to place some well-preserved examples belonging to different genera in my hands for investigation. Of all the specimens which have been sent to me, some fragments of Euplectella aspergillum in absolute alcohol are much the best preserved, and therefore the best suited for thorough examination. I commence with the description of the soft parts of this well-known and beautiful form, following the classification which has been adopted by Sir Wyville Thomson in his descriptions of the species.

The following paper contains a short digest of investigations communicated to the Society on several occasions during the past, and the present, session. The work had been for some months laid aside, but my attention was recalled to it by Professor Chrystal's valuable paper, in which he treats Minding's Theorem as an example of Plücker's methods, and also by the help of Rodrigues' co-ordinates. I am induced to publish a few of my results in full, as I think that a comparison of the analysis employed by Chrystal, with the very different analysis employed by myself, may be useful as well as interesting, especially from the point of view of the simplicity of the quaternion method. Even when the quaternion processes are written out at full length, they are in general shorter than the most condensed forms of ordinary analysis; and there can be no doubt that they are much more easily interpretable into the corresponding geometrical ideas.