M. Lagrange, on applying the method of continued fractions to the resolution of numerical equations, discovered that, for those of the second degree, the quotients recur periodically. From this, combined with the previously well known fact that all periodic chain fractions belong to quadratics, he inferred that periodicity is exclusively confined to equations of this order.

In January 1858, I showed to the Royal Society that the series of approximating fractions obtained by M. Lagrange can be continued in the opposite direction, and that the convergence then is to the other root; and enunciated the general theorem, that if any two fractions be assumed, and if a progression be formed from them by combining fixed multiples of their members, this progression, which I called duserr or two-headed, may be continued in either way, and gives on the one hand the one, on the other hand the other root of a quadratic.

The state of the vegetation in the open ground of the Botanic Garden during the month of December 1863 was so very remarkable that I have been induced to submit a notice of it to the Royal Society. The number of phanerogamous species and varieties in flower during the month amounted to 245; of these 35 were spring-flowering plants which had anticipated their period of florescence, while the rest were summer and autumn flowers which had protracted their flowering beyond their usual limits.

In the “Philosophical Transactions” for 1819, the author had shown that the doubly refracting force of calcareous spar extended beyond the sphere of the reflecting force, producing a change in the polarising angle varying with the inclination of the incident ray to the axis of the crystal, and producing a deviation of the plane of polarisation from the plane of incidence and reflexion, when the reflecting force of the crystal was reduced by contact with oil of cassia and other oils. These experiments were made on the face of the primitive rhomb.

It was shown by this paper that the gases dissolved in American petroleum, and which gave to it such a high degree of inflammability, were composed of the lower members of the marsh gas series, having the general formula,

and to which the liquid products have already been referred.

The gases evolved from the Pennsylvanian oil were collected at a temperature of −1° Cent., as they floated, mixed with air, over the surface of the liquid in the casks in which it is imported into this country and the hydrocarbons were shown by eudiometrical analysis to have the composition of a mixture in nearly equal proportions of the hydrides of ethyl and propyl.

More than a year since my attention was directed to a statement of Hofmann's, that the action of terchloride of phosphorus on aniline yielded a white substance of crystalline character, as the investigation of this compound so produced seemed likely to be of interest, I began its examination, but circumstances prevented me from completing it at that time. The publication of Schiff's papers, however, on the metal-anilides again drew my notice to the subject, and I considered it a duty to myself to publish the results of my experiments, more especially as the reaction which forms the subject of this paper appears to have escaped the attention of that chemist.

The author stated that in consequence of Legendre's work, the proof of Fermat's Theorem is reducible to showing the impossibility of

xm = ym+zm,

when m is an odd prime, x, y, z being integers.

Talbot has shown that in this case x, y, z are necessarily composite numbers.

The author shows, among other results of very elementary processes, that if numbers can be found to satisfy the above equation, x and y leave the remainder 1 when divided by m; and that z has m as a factor. Many farther limitations are given on possible values of x, y, z—the process being based on the consideration of their prime factors, and on Fermat's Elementary Theorem Nm − N = Nm.

The eye-pieces and object-glasses, each in a separate tube, worked in a case 15½ inches in length by 5 inches in breadth, and 2 in depth, forming, with the bevelled corners, a flat octagon, covered with a species of shagreen, and mounted in silver. An exceedingly simple and ingenious arrangement, consisting of a double screw working four small arms of brass, was placed at each end for the purpose of regulating the distances between each pair of glasses, and silver dial plates enabled the operator to set the instrument according to his ascertained requirements. The instrument belongs to the Royal Institution of Liverpool, and is supposed tc have been part of a collection of rarities, made by Wm. Roscoe, in Italy. As a telescope, it is of great power; the focus is adjusted by one portion of the case acting as a draw-tube within the other part.

After some preliminary remarks on the common origin of the Indians, Greeks, and Romans,—on the expectation thereby raised that we should find in the earliest literatures of these nations some remains of the primeval mythology which their ancestors must originally have possessed in common,—on the partial fulfilment of this expectation by an examination of these literatures,—and on the greater light thrown by the Rigveda than by any other monument of ancient poetry on the genesis of mythology,—the author adverts to the various theories of creation which would naturally be formed by simple men in the earlier ages of the world, to the manner in which the various great phenomena of nature would come to be ascribed to different deities, and to the diverse aspects in which the grander objects of creation, such as heaven and earth, were viewed, sometimes as inanimate, sometimes as animated and divine.

Having observed a series of serrated fringes in examining the colours produced by the fibres of the crystalline lens of fishes, the author was led to imitate them by the combination of grooves upon glass and steel surfaces, or of grooves taken from these surfaces upon isinglass or gums.

The interference bands thus produced were serrated or rectilineal, sometimes parallel and sometimes at right angles to the direction of the grooves, and varying in their magnitude and character according as they were exhibited on the colourless image, or on the diffracted spectra, or as they were produced at different angles of incidence, or at different distances of the grooved surfaces, or by different numbers of reflexion, or by different numbers and combinations of refracting and reflecting surfaces.

The subject of this essay may be described as an application of the Theory of Number to Geometry, its principal or characteristic problem being to determine under what conditions the sides or surfaces of figures may be represented by integer numbers.

Like all other inquiries into the properties of integers, it is rather speculative than practical, and yet, perhaps on that very account, is more apt to engross the attention of its cultivators than almost any other department of pure mathematics. It seems, indeed to be of very little moment whether we can demonstrate that the sum of two cubes never can be a cube; and yet one after another of the most eminent mathematicians have tried, and, though foiled, have again and again essayed the proof of Fermat's negation.

This is an abstract of an investigation which will be published in full in “Thomson and Tait's Natural Philosophy.” It contains the application of Green's wonderful results regarding the potential to the determination of the centre of gravity of a system when there is such a point. Some of the more remarkable propositions, which are thus established are as follows:—

If the action of terrestrial or other gravity on a rigid body is reducible to a single force in a line passing always through one point fixed relatively to the body, whatever be its position relatively to the earth or other attracting mass, that point is called its centre of gravity, and the body is called a centrobaric body.

The writer begins his communication by giving a brief description of the structures in which valves are found, and takes occasion to comment on the nature and properties of the veins and arteries, and on the arrangement of the muscular fibres in the ventricles, as these necessarily influence, to a greater or less extent, the action of the valves. He also adverts at some length to the shape of the venous, arterial, and auriculo-ventricular orifices, and to the fibro-cartilaginous rings by which the latter are surrounded; as well as to the dilatations or sinuses which are found behind or to the outside of the segments constituting the semilunar valves in the veins and arteries, and to the shape of the ventricular cavities, which, as he points out, bear an important relation to the valves, inasmuch as they determine the direction in which the blood acts upon them ; precise information on these points being, according to the author, indispensable to a just appreciation of the subject under investigation.

This paper was an attempt to submit to a severe and searching examination, the very new and apparently important ideas contained in the work, published four years ago by Mr John Taylor of London, and entitled “The Great Pyramid; why was it Built?” To this end, the original authorities for measures of the Pyramid, had been extensively referred to, from Professor John Greaves in the 17th century, down to Colonel Howard Vyse and Dr Lepsius in the 19th; and their various and sometimes conflicting numerical statements had been computed with all due attention to scientific accuracy, as well as every endeavour to eliminate both personal and other sources of error in tbe observations.

The author having had experience of the simple duodecimal monetary system of Portugal, which is accumulated by multiplication from a decimal minimum unit, proposes to apply the same principle to the British coinage. Having also discovered, while in Canada, that the difference betwixt the British coinage and a decimal position or character, is 20 per cent, over all, he proposes to overcome this by altering the value of the present farthing, by this rate, making it the fifth, instead of the fourth of the penny—the penny of 5 farthings would then become the duodecimal multiplier of all the coins, making the shilling 60 farthings instead of 48, and the pound 1200 instead of 960, and each intermediate coin would be represented by one of the digits and one or two ciphers, while the pound, being represented by 12, the highest single divisor or multiplier, could be kept out as the maximum unit.

Peclet found, by his own experiments, that a body with any common unpolished non-metallic surface, kept by heat from within at 1° higher temperature than that of the air and other objects round it, loses heat from each square metre of surface at the rate of about nine kilogramme-water thermal units per hour, or, which is the same, of a gramme-water unit from each square centimetre per second. The mean conductivity of the three Edinburgh strata, in which Principal Forbes's underground thermometers were placed, is 2¾ grain-water units per second per square foot per 1° per foot rate of variation of temperature, as I have shown previously.

Mr Skene in his paper gave a sketch of the various views which had been entertained regarding the Celtic languages, and advocated the opinion that these languages belonged properly to the Indo-European class. He gave an outline of the distinctive peculiarities of the two branches of the Celtic languages, the Welsh and the Gaelic, and discussed the leading theories regarding the lost Pictish language, only five words of which were now extant. Mr Skene had, as the result of his investigations, arrived at the conclusion that the Pictish language occupied a place between the Welsh and the Gaelic—that it was a Gaelic dialect partaking largely of Welsh forms.

The effect of dew in protecting vegetation every clear still night of summer was long ago pointed out by Dr Wells; the correctness and acuteness of whose views on this subject have been generally recognised. The hypothesis recently put forth by Dr Tyndall, that absorption of radiant heat by aqueous vapour in the atmosphere is an effective defence against destructive degrees of cold, and the ready acceptance yielded to it by some of our highest authorities in the popular promulgation of the truths of science, seem to render it necessary to recall attention to Dr Wells' admirable work.

In this paper there is presented a series of characteristic examples of vital agency, followed by some considerations bearing on the extent to which, in a physiological point of view, it can be admitted that the physical forces, under exchangeable forms, are concerned in organic phenomena. At the same time, one of the particular objects of inquiry is, whether there be ground for the belief that vital agency, being not a force but a directive principle, may accomplish all the parts assigned to it, solely by means of the physical forces variously modified through organic structure.

The author was led to examine the sun pictures taken by the Kew Photoheliograph, with the view of ascertaining if any connection exists between the behaviour of sun spots and planetary configurations.

It was found, that when any portion of the sun's disc recedes by virtue of rotation from the neighbourhood of Venus, it acquires a tendency to break out into spots, and, on the other hand, when such approaches Venus, there is a tendency towards the healing up of spots. Carrington's observations were then discussed, which seem to show that, on the whole, the sun's surface is fullest of spots when Jupiter is furthest from our luminary, and freest from spots when he is nearest.