The forces concerned in the laying and lifting of deep submarine cables attracted much public attention in the years 1857–58.

An experimental trip to the Bay of Biscay in May 1858, proved the possibility, not only of safely laying such a rope as the old Atlantic cable in very deep water, but of lifting it from the bottom without fracture. The speaker had witnessed the almost incredible feat of lifting up a considerable length of that slight and seemingly fragile thread from a depth of nearly 2½ nautical miles.

This paper commences with a condensed re-statement of the fundamental principles and formulæ of the Dynamical Theory of Heat, from the first six parts of the author's treatment of the subject previously communicated to the Royal Society of Edinburgh, and his articles “On the Thermo-elastic Properties of Matter,” in the “Quarterly Mathematical Journal” (April 1855), and on “Thermo-magnetism,” and “Thermo-electricity,” in Nichol's Cyclopedia (Edinburgh 1860).

The “Doctrine of Uniformity” in Geology, as held by many of the most eminent of British Geologists, assumes that the earth's surface and upper crust have been nearly as they areat present in temperature, and other physical qualities, during millions of millions of years. But the heat which we know, by observation, to be now conducted out of the earth yearly is so great, that if this action had been going on with any approach to uniformity for 20,000 million years, the amount of heat lost out of the earth would have been about as much as would heat, by 100° Cent., a quantity of ordinary surface rock of 100 times the earth's bulk.

We now know a considerable number of elements which exhibit, in their compounds, two or more distinct degrees of atomicity. Thus we have N triatomic in ammonia and its analogues, in nitrous acid, the nitrites and the nitroso-substitution products; pentatomic in the ammonia salts, in nitric acid, and the nitro-substitution products. P triatomic in PH3, PCl3, phosphorous acid, and the phosphites; pentatomic in PCI5, POCI3, H3PO4, HPO3, &c.; C diatomic in CO, and tetratomic in almost all other compounds; a large number of such examples might be adduced, but these may suffice.

In Volume XXI. of the Transactions of the Royal Society of Edinburgh, pages 407–409, a proposition in the Theory of Numbers is demonstrated. The proposition may be extended, and, at the same time, the demonstration simplified. I propose to establish the following result:—Let 1, α, β, γ, … μ, be the n roots of the equation

then, will

where t is any quantity.

In a former paper on this subject, the author gave a selection from his observations on the polarisation of the atmosphere. After its publication, he received a long and elaborate memoir on the same subject by Dr R. Rubenson, containing a series of valuable observations made at Rome on the place of maximum polarisation, and on the intensity of the maximum polarisation at different hours of the day. Important observations on the polarisation of the atmosphere have also been made by M. Liais and M. Andres Poey, and the value of such observations, in determining the height and constitution of our atmosphere, has been universally recognised.

The county of Fife abounds in caves or “weems”—a derivative from the Gaelic name for caves—and their existence gives a title to the earldom of Wemyss. Some of the caves in Fife are historical, as St Rule's at St Andrews, St Adrian's near Elie, and St Margaret's at Dunfermline. St Serf of Culross the great patron saint of the west of Fife, is described by one of his biographers as having usually spent the forty days of Lent in a cave named, as such retreats often were, the Desertum. This cave at the Desertum— (or Dysart, to use the modern form of the name)—was used as a church up till near the time of the Reformation. About two miles eastward of Dysart, and near the village of Easter Wemyss, there is a range of large caves, seven or eight of which are at the present time open; but several more probably exist, having their openings covered over with debris. They stand about 15 or 20 feet above the level of high tide. Some of them are 80 to 100 feet in length, and of corresponding height and breadth.

The paper consists mainly of the author's observations on the microscopic anatomy of the reproductive organs (with illustrative figures) of the following new species and varieties of Lichens and Fungi collected by him in the province of Otago, New Zealand, in 1861:—

In the “Flora Novæ-Zelandiæ” of Dr Hooker there is no record of any species of Opegrapha, Platygrapha, Umbilicaria, or Abrothallus; and Professor Churchill Babington therein specially remarks, indeed, on the supposed absence of the genera Opegrapha and Umbilicaria (Gyrophora) from the Lichen-Flora of New Zealand.

The Council reported that they had awarded the Keith Prize for the biennial period 1863–65, to Principal FORBES, St Andrews, for his “Experimental Inquiry into the Laws of Conduction of Heat in Iron Bars” which was printed in the last Part of the Transactions of the Society.

The Council also reported that they had awarded the Neill Prize for the triennial period 1862–65 to Andrew Crombie Ramsay, F.R.S., Professor of Geology in the Government School of Mines, and Local Director of the Geological Survey of Great Britain, for his various works and memoirs published during the last five years, in which he has applied the large experience acquired by him in the Direction of the arduous work of the Geological Survey of Great Britain to the elucidation of important questions bearing on Geological Science.

In the course of the detailed investigations which, during the past six or seven years, have been carried on by the officers of the Geological Survey into the history of the glacial period in Britain, the desire naturally arose to compare the phenomena of glaciation so familiar in this country with those of some other region where they might be linked on to the action of still existing glaciers. No other part of Europe offered so many facilities for such a comparison as were to be found in Scandinavia. It was accordingly planned by my colleague, Dr John Young, and myself, to visit Norway in the summer of 1863. Unforeseen circumstances delayed the journey, and ultimately deprived me of the companionship of my friend.

The object of the paper was to point out that the theory of variables has a third branch, bearing to the Differential and Integral Calculus a relation somewhat analogous to that which the theory of Logarithms bears to Involution and Evolution.

In the theory of Functions there are three connected variable quantities, viz., the Primary, the Function, and the Derivative. When the relation between the primary and the function is given, the discovery of the derivative belongs to the Differential Calculus. When that between the primary and the derivative is given, the function is found by help of the Integral Calculus. And the general problem to find the primary when the relation between the function and its derivative is given, belongs to a third branch, which may be called the Calculus of Primaries.

This paper was a sequel of a paper read to the Royal Society, and published in the 23d volume of the “Transactions.” That paper was entitled, “On the Variations of the Fertility and Fecundity of Women, according to age;” and among other conclusions therein announced, was one to the effect that fecundity, or likelihood of child-bearing after marriage, increased from the earliest child-bearing age till about twenty-five, and thereafter gradually decreased. In the present paper, the laws regulating such fertility after marriage are demonstrated. The chief data on which the arguments for these laws are based are derived from an analysis of 16,301 families, in connection with which entries were made in 1855 in the public registers for Edinburgh and Glasgow.

In this paper the author proposes to use for purposes of classification those radicals or parts of molecules which are common to genera of substances, and within which the changes characteristic of these genera take place.

In an air-bubble of moderate dimensions the pressure is very little greater than that of the atmosphere; but, as the difference of pressures within and without a bubble is proportional to its curvature, it appeared to me possible that the so-called solution of air in water might be due in some degree to the extreme compression of the air when divided into small bubbles invisible perhaps even under the microscope.

In “Tait and Steele's Dynamics of a Particle,” chap, iv., a number of geometrical constructions are given, some possibly for the first time, connected with the motion of projectiles in parabolic paths in vacuo. I have recently been led to remark that most of these propositions still hold when we substitute, for the uniform action of gravity in parallel lines, the action of gravity supposed to be directed to the earth's centre, and to vary inversely as the square of the distance from that point. My excuse for bringing so simple a matter before the Society is; that the propositions are in themselves curious and elegant, and that I am not aware of their having been before mentioned. A very few examples will suffice to indicate the change of form required by the more general assumed conditions.

On the banks of the Elwand Water, which runs into the Tweed, about two miles above Melrose, there is a picturesque glen called the Fairy Dean, which has become a favourite place of resort, from its association with the incidents in “The Monastery” by Sir Walter Scott. It has acquired an interest of a different kind from certain mineral concretions which have received the name of Fairy Stones, from their being found in that part of the rivulet which runs through the Fairy Dean.

When the Waverley Novels were not acknowledged by their author, facts or incidents to which they referred, were always welcome subjects of conversation at Abbotsford; and on one occasion when I happened to mention that singular stones were found in the Fairy Dean, Sir Walter Scott expressed a desire to see them, and to know how they were formed.

In 1823 the Royal Society established a register of hourly observations of the thermometer at Leith Fort. They were made by the non-commissioned officers of artillery, and were continued for four years, from 1824 to 1827 inclusive. A report on these observations for the years 1824 and 1825 was published in the tenth volume of the Society's Transactions; but from causes to which it is unnecessary to refer, the report on the observations of 1826 and 1827 were not then published.

In a paper on hemiopsy, the author had shown that the parts of the retina affected with this disease were susceptible of luminous impressions and insensible to visual ones, and that the light by which they were impressed was derived by irradiation from the adjacent parts of the retina. The parts of the retina affected with hemiopsy were, however, so small, so irregularly distributed, and the phenomenon of such short duration, that it was difficult to study it and deduce any satisfactory results.

In this paper absolute sterility is held to mean the condition of a woman who, under ordinary favourable circumstances for breeding, produces no living or dead child, nor any kind of abortion. Sterility is held to mean the condition of a woman who, under ordinary favourable circumstances for breeding, produces no living and viable child, or adds not one to the population. Relative sterility is held to mean the condition of a woman who, while she may or may not be absolutely sterile, while she may or may not be sterile, is, under ordinary favourable conditions for breeding, sterile in relation to the circumstance of time, or, in other words, in relation to her age and the duration of her married life.

In cases of cleft palate with alveolar fissure, the maxillary bones are, during infancy, not only ununited, but, in general, if not always, more widely separated from one another than in the natural condition. This has been frequently observed in such cases as come under the care of the surgeon, although little attention appears to have been bestowed upon the fact beyond its mere casual mention. Measurements, however, have been lately made by Dr Engel, showing that the difference between certain fixed points—such as the two infra-orbital foramina, the nasal processes of the upper jaw, &c, is very well marked, when the distance is measured in a healthy new-born child, compared with one having a cleft palate.