The fact that certain bodies expand on solidifying, as in the case of water, has long been well known, and this property has been recognised in some of the metals, owing to their filling the mould in which they are cast so as to reproduce the finest lines. The fact of their so doing is, however, only known qualitatively—no accurate measurements having, so far as we are aware, been recorded. The property being of great interest to ironfounders, we have undertaken a series of experiments to determine its real value—the materials being put at our disposal by Messrs M'Dowall, Steven, & Co., to whom we tender our best thanks.

In these experiments the general method pursued was by direct contact and separation of two circular plane metal disks, the lower one of which was insulated and connected to one pair of quadrants of a Thomson quadrant electrometer. The upper disk or plate of this condenser arrangement pressed during contact on the lower by its own weight, and was in connection with the other pair of quadrants and with the earth. The lower plate formed the upper surface of a cylindrical flask, whose temperature was determined by that of the water contained within it.

In 1877, shortly after the disastrous colliery explosion in the Blantyre pit, in which hundreds of lives were lost, Mr James Young, F.R.S., of Kelly, described to me an instrument which he had thought of for determining what is the amount of fire-damp in any part of a mine. This was the first thing which directed my attention to the subject, and I very soon saw that there was a principle in acoustics which might be most admirably adapted to the end in view, viz., to determine the quantity of fire-damp (or marsh gas) by the diminution in density of the mixed air and gas (for marsh gas is only about half the density of air). Mr Young and myself tested the principle the next day, and found it to be one of extreme delicacy.

The author commenced by stating that he had been led to make farther investigations, on the subject of his hypothesis as to the nature of comets, by some comparatively recent criticism to which that hypothesis had been subjected. Its main features had been published more than ten years ago in the “Proceedings” of the Society (May 17, 1869) and in the first volume of “Nature.” Of course, if a critic completely misstates an hypothesis, he has no difficulty in refuting it; so that to such writers the author does not attempt to reply.

In a paper read to the Society in June last, “Proceedings,” June 1879, on the fungus disease affecting salmon and other fish, I discussed the various theories which were advocated, as to the cause of the disease; the effects of the fungus upon the fish, the vegetative and reproductive aspects of the fungus, and the belief that salt water had a curative effect upon salmon affected with fungus disease, on their reaching and remaining for some time in that element.

The survey which forms the subject of the present paper extends over an area of 6000 square miles, bounded by the Jordan on the east, the Mediterranean on the west, the river Leontes and the springs of Jordan near Dan on the north, and the desert of Beersheba on the south. Within these limits a complete triangulation, with two bases each about four miles long, has been extended, and the whole of the country mapped to the scale of one inch to the mile. The work occupied five years in the field, and nearly three years more in preparing the results for publication.

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.

Besides Minding's original investigation, several others have been given since. The last of these, due to Professor Tait, rests on purely quaternion methods, and is so elegant and concise that I was led to reinvestigate the whole subject by ordinary methods in the hope that the analysis might have some points of interest.

While making some algebraic problems last summer for an examination, I devised the following question:—

“A schoolmaster went mad, and amused himself by arranging the boys. He turned the dux boy down one place, the new dux two places, the next three, and so on till every boy's place had been altered at least once. Then he began again, and so on; till, after 306 turnings down, all the boys got back to their original places. This disgusted him, and he kicked one boy out. Then he was amazed to find that he had to operate 1120 times before all got back to their original places. How many boys were in the class?”

This equation has been exhaustively treated in our Transactions by M. G. Plarr. The present note is a mere sketch of a graphical solution.