If U, V, &c, are rational and integral functions (٭)(x, y, z)r, all of the same degree r, in regard to the co-ordinates (x, y, z), then √U + √V + &c. is a polyzome, and the curve √U + √V + &c. = 0 a polyzomal curve. Each of the curves √U = 0, √V = 0, &c. (or say the curves U = 0, V = 0, &c.) is, on account of its relation of circumscription to the curve √U + √V + &c. = 0, considered as a girdle thereto (ζῶμα) and we have thence the term “zome” and the derived expressions “polyzome,” “zomal,” &c. If the number of the zomes √U, √V, &c. be = ν, then we have a ν-zome, and corresponding thereto a ν-zomal curve; the curves U = 0, V = 0, &c., are the zomal curves or zomals thereof.

Although it is very improbable that there remains to be discovered any new, and at the same time simple, fact connected with a question which has been elaborately treated by many of the greatest mathematicians of this and the preceding century, the employment of a new mathematical method may enable us to present some of their results in a more intelligible form, and with far less expenditure of analytical power than has hitherto been deemed necessary; and it may give us such an insight into the question, that we shall be able easily to discover the mutual relations among the various processes which have been already employed; so far, at least, as these differ in principle, and not merely in the peculiar co-ordinates assumed for the purpose of simplifying the equations. Such a method is that of Quaternions, which seems to be expressly fitted for the symmetrical evolution of truths which are usually obtained by the ordinary Cartesian methods only after great labour of calculation, and by modes of attack so indirect, and at first sight so purposeless, as to bewilder all but a very small class of readers. Quaternions afford so clear a view of the nature of the question they are applied to, that even the student, if he have some little knowledge of them, can often see why a transformation is made, whose object he would have been unable to discover had the problem been masked in the unnecessarily artificial difficulties of Cartesian geometry, or the outrageously repulsive formulæ of spherical trigonometry.

The anatomy of the soft worms variously arranged under the Nemertean Order has, even in recent times, not been carried out with that completeness necessary for their thorough elucidation, a state of matters partly due to the confounding of the structure of one family with another, and predicating of the series what investigation has but proved in one group. Few British comparative anatomists have paid much attention to these animals; indeed, Dr George Johnston, Mr Harry Goodsir and Dr Thomas Williams, are the only three who have left researches of any moment on the subject. The observations of the first-mentioned naturalist were made many years ago, with the aid of inferior instruments, and, though conscientious enough, are very meagre and unsatisfactory; and those of Dr Williams, while also showing the defects just noted, bear evident traces of imagination. Mr H. Goodsir's interpretation of structures was, from his limited observations, likewise very erroneous. On the Continent, again, the investigators have been more numerous, and a long list of distinguished names attest the interest which the subject has received at their hands.

I. On some Transformations in Films when brought in contact with Surfaces of Glass.

(1.) Let a film be formed on the rim of a cylindrical wine-glass, at or very near its margin; cover it immediately with a watch-glass, and holding the latter firmly in its place, invert the whole, so that the film is placed in a vertical position. The film will now attach itself to the watch-glass at the lowest point where it is in contact with the margin of the wine-glass, and will run up the concave surface of the former. At the same time, the film will leave the margin of the wine-glass at its upper edge, and retreat into the glass, running down its inner surface. A film of the form of the segment of a sphere will thus be produced, which, with the upper portions of the inner surfaces of the watch-glass and wine-glass, will form a hollow filled with air, as shown in fig. 1. This state of matters will remain the same, in whatever position we now place the wine-glass, the figure which has thus been produced being one of equilibrium. The phenomenon produced arises from the fact, that when the original film is first taken up on the margin of the wine-glass, a drop of liquid always remains in the bottom of the glass, and when the glass is inverted, so as to bring the film into the vertical position, this drop runs down the inner surface of the glass till, reaching its lowest point, it brings the film into close contact with the watch-glass; the film now attaches itself to and spreads itself over the surface of the latter in a similar way to what takes place when a bubble blown from a pipe is brought into contact with any smooth surface, whilst the original system of equilibrium being now disturbed, the upper part of the film is put in motion, and a new system of equilibrium is formed.

The observations which form the basis of this paper were made on newly-born children with the object of determining whether their temperature differed from that of the adult, and if so, how and to what extent. They were made during a recent residence in the Edinburgh Maternity Hospital, with the full sanction of the attending physician, Dr Charles Bell. The thermometers used were Casella's straight self-registering. The rectum was the part chosen for the insertion of the instrument. The utmost expedition was used in ligaturing the cord, and separating the child from its mother. This being effected, the bulb of the thermometer was at once introduced into the rectum, and the child was wrapped in flannel, and committed to the charge of a nurse, who held the instrument steadily in situ. In five minutes it was removed, and the temperature noted. The observations were repeated every fifteen minutes during the first hour, every thirty minutes during the second and third hours, and then every hour up till the sixth hour after birth; after that at wider intervals up to twenty-four hours; and then only twice a day between ten and eleven in the morning, and between six and seven in the evening.

During the last three years I have made a large number of observations on the temperature of the common fowl under different circumstances, the results of which I now beg leave to submit to the Society, with the hope that they may be considered not altogether uninteresting to the physiologist.

The fowls tried were chiefly of the pure Dorking breed. At the time they appeared to be healthy, and all in good condition. They had all the run of a field adjoining the poultry yard.

In all the trials the same thermometer was used,—each degree of which, that of Fahrenheit, was divided into ten parts, and had been warranted correct by the makers after comparison with a standard.

As the reputed fact of Archimedes having burned the Roman ships engaged in the siege of Syracuse, by concentrating on them the solar rays, has not only been doubted but disbelieved by some of the most eminent scientific men, I shall briefly give the evidence on both sides.

The burning of the ships of Marcellus is mentioned by most of the ancient writers who refer to the machines which Archimedes employed in the defence of his native city, and their statements have been repeated by succeeding authors, without any doubts having been expressed until comparatively recent times. Our earliest authorities on the subject are Diodorus Siculus, Lucian, Galen, Dion, and Pappus.

The theory of reciprocal figures used as diagrams of forces was first completely stated by Professor T. Clerk Maxwell, in a paper published in the “Philosophical Magazine,” April 1864. The following definition of reciprocal plane figures, and their application to statics, are there given as follows:—

“Two plane figures are reciprocal when they consist of an equal number of lines, so that corresponding lines in the two figures are parallel, and corresponding lines which converge to a point in one figure form a closed polygon in the other.”

“If forces represented in magnitude by two lines of a figure be made to act between the extremities of the corresponding lines of the reciprocal figure, then the points of the reciprocal figure will all be in equilibrium under the action of these forces.”

The demonstration of this statement is given. The conditions under which stresses are determinate, and some examples of reciprocal figures, are also given in the paper, which leaves nothing to be desired by the mathematician.

There can be no reasonable doubt that a relation exists between the physiological action of a substance and its chemical composition and constitution, understanding by the latter term the mutual relations of the atoms in the substance. There are numerous indications of such a relation, and attempts have been made to express it formally in certain cases. Thus it has been long observed, that the salts of the same base have a common physiological action, and it has been pointed out by Mr Blake that, with some exceptions, the salts of isomorphous bases have a similar action. A corresponding likeness in physiological action may be traced in salts having the same acid, but beyond these generalisations we are not aware that any approach has been made to the statement of a law connecting the physiological action of a substance with its chemical constitution.

Authorities on the action of medicinal substances agree in including convulsions among the effects on man of belladonna, and of its active principle, atropia. Similar effects are described as occurring when large doses of these substances are administered to dogs, rabbits, and other mammals, and to various birds. The recent remarkable progress of our knowledge of the exact and ultimate physiological action of many medicinal substances is greatly due to investigations that have been made on animals of a lower type of organisation; and, accordingly, numerous observers have instituted experiments with atropia on such animals, and especially on frogs. Hitherto, however, convulsions and tetanus have not been described among the effects of atropia-poisoning in cold-blooded animals.

It is now many years since Dr Whewell drew the attention of the Cambridge Philosophical Society to the courageous, if somewhat Quixotic, attempts of Hegel to cast discredit on Newton's law of gravitation, and on the mathematical demonstrations of Kepler's laws given in the “Principia.” At the time when Whewell wrote, it would probably have been difficult to find in Britain any one ready to maintain the cause of Hegel in this matter, or even to hint that the astounding arguments of the Naturphilosophie flowed from any deeper source than self-complacent ignorance.

The present state of matters is different. The philosophy of Hegel is now for the first time beginning to have a direct and powerful influence on British speculation. Men are beginning to study Hegel; and an author whose works confessedly demand the labour of years, if they are to be fully understood, can hardly be studied at all except by devoted disciples. A man whose determination to master Hegel's philosophy survives the repelling impression which the obscurity and arrogance of the philosopher are sure to produce at first, is very likely to be carried away by the calm assumption of omniscience which runs through Hegel's writings. It is not, therefore, surprising that Dr Stirling extends his admiration to Hegel's physical positions; and if he does not venture to say that Hegel's proof of Kepler's laws is right, at least feels sure that it would repay the attention of mathematicians.

In the fourth part of these researches, I described a new base produced by the action of sodium upon picoline, to which I gave the name of Parapicoline, because it has the same composition as picoline, although the circumstances of its formation appeared to show that it had been produced by the combination of two molecules of that substance, so that its true formula would be C12H14N2. Unfortunately, its high boiling point, and tendency to decompose when distilled, made it impossible to determine its vapour density, which afforded the only means of ascertaining whether this hypothesis was correct; and it was only assumed, because similar cases of polymerisation had been established beyond a doubt in the case of other classes of organic compounds. In the hope of obtaining a similar base of lower boiling point, and therefore better adapted to the necessary experiments, I have submitted pyridine to the action of sodium, and the results of the inquiry are contained in the following pages.

1. The mathematical work of the present paper has been performed to illustrate the hypothesis, that space is continuously occupied by an incompressible frictionless liquid acted on by no force, and that material phenomena of every kind depend solely on motions created in this liquid. But I take, in the first place, as subject of investigation, a finite mass of incompressible frictionless fluid completely enclosed in a rigid fixed boundary.

In the course of my studies on the Microscopic Anatomy of Lichens, during the last fifteen years, I have frequently met with various more or less minute Parasites—mostly black and punctiform or papillæform—sometimes disciform or maculæform—affecting either the thallus or apothecia of lichens, or both thallus and apothecia. They grow equally on foliaceous and crustaceous lichens, especially of the following genera:—Parmelia; Physcia; Umbilicaria; Solorina; Peltidea; Nephromium; Sticta; Stereocaulon; Usnea; Neuropogon; Cladonia; Bæomyces; Squamaria; Placodium; Lecanora; Pertusaria; Thelotrema; Lecidea; Graphis; Endocarpon; Verrucaria.

§ 1. Object of this Paper.—In a paper on the Mechanical Action of Heat, which I sent to the Royal Society of Edinburgh in December 1849, and which was read in February 1850, it was shown, that if sensible or thermometric heat consists in the motion of molecular vortices supposed to be arranged in a particular way, and combined in a particular way with oscillatory movements, the principles of thermodynamics, and various relations between heat and elasticity, are arrived at by applying the laws of dynamics to that hypothesis. The object of the present paper is to show how the general equation of thermodynamics, and other propositions, are deduced from the hypothesis of molecular vortices, when freed from all special suppositions as to the figure and arrangement of the vortices, and the properties of the matter that moves in them, and reduced to the following form:—That thermometric heat consists in a motion of the particles of bodies in circulating streams, with a velocity either constant or fluctuating periodically. This, of course, implies that the forces acting amongst those particles are capable of transmitting that motion.

In a paper read before the Royal Society of Edinburgh in April 1843, Dr Maclagan described the general properties of the alkaloid, whose presence had been indicated in the bark of the bebeeru or greenheart tree, by Dr Rodie of Demerara, and described the mode of preparation of its sulphate for medicinal use. The fact that bebeerine appeared to possess marked antiperiodic properties, rendered its careful chemical study desirable, and accordingly the alkaloid, purified as far as possible, was subjected to analysis by Drs Maclagan and Tilley. It resulted from this research that bebeerine is an uncrystallisable base, very soluble in alcohol, less so in ether, and very sparingly so in water. It forms with acids salts which are all uncrystallisable. With perchloride of gold, mercury, copper, and platinum, it gives precipitates which are soluble to a certain extent in water and alcohol, but which are deposited in a non-crystalline form when the solution cools. To this base the author assigned the formula C35H20N2O6(C = 6). Von Planta subsequently attempts to purify further the alkaloid, and assigned to it the formula C36H21O6N(C = 6) or C18H21O3N(C = 12.)

Charts, showing by Isobaric Lines the mean pressure of the atmosphere over the globe during the months of the year, may be justly regarded as furnishing the key to all questions of meteorological inquiry; for without the information conveyed by such charts it is impossible to discuss satisfactorily those questions which relate to prevailing winds, the varying temperature, and the rainfall throughout the year in the different countries of the world. It is to meet this desideratum that the Charts of Mean Atmospheric Pressure of the globe which are given with this paper are offered as the first approximate solution of this great physical problem.

Since Part I. was read in March 1868, valuable additional information has been obtained from Australia, New Zealand, Tasmania, Africa, South America, the west coast of North America, Iceland, Norway, and Sweden, and from several isolated stations in different parts of Europe and Asia. The period for the British Islands and a large portion of Europe has been extended so as to include the eleven years from 1857 to 1867.