The sines of the successive equidiiferent arcs form a progression having for its general character the relation

and the properties of sines may be deduced from this general formula. Viewed in this light, the angular functions become cases only of more general ones.

As we cannot have knots in two dimensions, and as Prof. Klein has proved that they cannot exist in space of four dimensions, it would appear that the investigation of their properties belongs to that class of problems for which the methods of quaternions were specially devised. The equation

where ϕ is a periodic function, of course represents any endless curve whatever. Now the only condition to which variations of this function (looked on as corresponding to deformations of the knot) is subject, is that no two values of ρ shall ever be equal even at a stage of the deformation. Subject to this proviso, ϕ may suffer any changes whatever—retaining of course its periodicity. Some of the simpler results of a study of this novel problem in the theory of equations were given,—among others the complete representation of any knot whatever by three closed plane curves, non-autotomic and (if required) non-intersecting.

Two earthquake shocks have lately occurred in Argyleshire of so decided a character that a description of their effects, as observed at four of the Lighthouse stations on the west coast of Scotland, will, it is thought, be interesting to the Society.

The first shock occurred on the 11th March, and was observed at the Lighthouse station of Hynish in the island of Tyree, and at Sound of Mull, near Tobermory, the distance between the two places being about 34 statute miles.

The author, in laying before the Society a revised and condensed version of the various papers recently communicated by him, took occasion to make some additional remarks. Of these only one need be given here. He pointed out that another fundamental term is requisite besides those already used viz., Knots and Links. For three endless cords may be inseparably entangled with one another, or locked together, even if no one of them be knotted and no two interlinked.

The “Challenger” Expedition had no opportunity of visiting Barbadoes, and this I regretted greatly, as Sir Rawson Rawson, who was at that time governor of the island, had paid great attention to the marine fauna, and was anxious to introduce us to his fine collection, which included many specimens of the rare and remarkable forms for which the sea of the Antilles is famous.

In this communication the author stated that he limited his remarks on the present occasion to some of the more prominent results he has arrived at in the course of this investigation. The paper itself, with the tables, will be submitted when the computations have been finished and thoroughly revised,—a work which must necessarily yet take some considerable time.

By means of a very delicate galvanometer, transient currents were detected when one of two plates of the same, or of different, metals, separated by a sheet of mica or glass, was suddenly heated.

The barometer only indicates real pressure when the atmosphere is in a state of perfect rest. It may then be represented a existing in vertical columns, but when it moves over a resisting surface its lower surface currents will be greatly retarded, while those aloft will move comparatively free and unimpeded. In this state it may be represented as moving in columns inclined in the direction towards which it moves. The atmosphere may thus be conceived as being divided into a number of spheroidal concentric layers, each of which is possessed of a different rate of speed, and moves more rapidly than the one beneath it: an increasing amount of friction will take place betwixt the layers as they approach the surface where its influence is greatest.

Accident has recently called my attention to a work entitled Essays on Scientific Subjects, by Robert Blair, Regius Professor of Practical Astronomy in the University of Edin. (Edin. 1818). In the University Library there is a second edition of a part of the same work with the title Scientific Aphorisms (Edin. 1827). I bring them before the notice of the Society, as they contain an explanation of gravitation, &c., almost identical with that of Le Sage, to which our attention was lately recalled by our President. Professor Blair seems to have invented this explanation for himself—because, though he gives frequent references to other authors, whose results he quotes, he makes, so far as I have seen, no reference to Le Sage.

Whatever be p and q it is obvious that

Hence

and so on. Finally we see that

absolutely without any restriction on the values of the quantities involved.

As we could not find pure cobalt for sale in the state of foil or wire suitable for thermo-electric experiments, we resolved to prepare it by electrolysis of chloride of cobalt in the way recommended by Mr G. W. Beardslee of Brooklyn, N.Y. For this ther e is require d pur e chlorid e of cobalt of a certain strength to form the electrolyte, and spongy metallic cobalt to form the positive electrode. By the kindness of Dr Crum Brown these were prepared with very great care in the chemical laboratory by Mr Robinson, to whom we are much indebted. The following note of the method of preparation employed has been supplied by Mr Robinson.

On March 25th, at 8.15 P.M., very few clouds being visible, there appeared a luminous arch in a vertical plane, passing true E. and W. nearly through the zenith. This arch, to my surprise, gave everywhere a continuous spectrum. I had, in company with Professor Dewar, observed a phenomenon similar in all respects (continuous spectrum included) some seven years before, and was then inclined to fancy the arch a mere low-lying cloud, and its light due to the town lamps. But on the present occasion, after about forty minutes, when the arch had disappeared, an aurora, showing unmistakably in the spectroscope the characteristic green band, appeared near the magnetic north. And a few days afterwards there appeared in the Edinburgh newspapers several descriptions of a similar arch seen at almost exactly the same time at Colinton, &c., and even so far off as Dunfermline. Most of the observers remarked the entire absence of coruscations as a proof that the phenomenon could not be auroral. The facts stated seem, however, to leave no doubt that it was connected with the aurora and the peculiarity of its spectrum remains to be explained.

Elonichthys ovatus, sp. nov. Traquair.

Of this I have only seen one specimen, from the limestone of Burdiehouse, and preserved in the Edinburgh Museum of Science and Art.

Description.—Allowing for the anterior part of the head, which is deficient, the entire length of the specimen to the extreme point of the upper lobe of the caudal fin would be about 5⅝ inches; the greatest depth of the body in front of the dorsal fin is 1½ inch. The distance from the origin of the pectoral fin to that of the ventral is a little over 1 inch, to opposite the commencement of the dorsal 1¼ inch, to opposite that of the anal 1½ inch, and to opposite that of the lower lobe of the caudal nearly 3 inches. The general form of the fish is thus remarkably short, deep, and ovoid, and its general appearance does not indicate that its peculiar form is due to post mortem distortion or change.

This paper contains an account of the cranial structure of Rhizodopsis as far as this can be elucidated by an examination of a suite of specimens from the Coal Measures of Fenton, Staffordshire, in the collection of Mr Ward of Longton, supplemented by additional specimens from various other localities.