Introductory Remarks.—The principles set forth in this paper, though now (with the exception of the first theorem) published for the first time, were communicated to the French Academy of Sciences fifteen years ago, in a memoir entitled “ De l'Equilibre intérieur d'un Corps solide, élastique, et homogène,” and marked with the motto, “ Obvia conspicimus, nubem pellente Mathesi,” the receipt of which is acknowledged in the Comptes Rendus of the 6th April 1857.

(1.) Principle of Isorrhopic Axes.—The following theorem was first published in the “Philosophical Magazine” for December 1855.

Prop. I. “Theorem. Every self-balanced system of forces applied to a connected system of points is capable of resolution into three rectangular systems of parallel self-balanced forces applied to the same points.

Two figures are reciprocal when the properties of the first relative to the second are the same as those of the second relative to the first. Several kinds of reciprocity are known to mathematicians, and the theories of Inverse Figures and of Polar Reciprocals have been developed at great length, and have led to remarkable results. I propose to investigate a different kind of geometrical reciprocity, which is also capable of considerable development, and can be applied to the solution of mechanical problems.

At least two species of Tubifex are abundant in Scotland, one of which is, perhaps, more prevalent in streams, the other in lakes. The former is common at the margin of the river Tay, when the water is low, in sandy tubes of little tenacity, and often in tunnels formed amongst the wet sand under stones, just as many of the marine annelids occur. Its length varies from three-fourths of an inch to an inch and a-half when stretched, and the segments range from fifty to seventy, the average number probably being sixty. The body is of various shades of dull fawn or pale madder-brown, somewhat interrupted by the pale bands at the junctions of the segments, and streaked by the reddish dorsal vessel; or, in summer, marked anteriorly by the whitish mass of the reproductive organs. It is easily recognised amongst its fellows by its faintly purplish or lilac hue, as well as by its peculiar mode of progression; and not a few are observed with reproducing heads and tails. This form, perhaps, has most claim to be called Tubifex rivulorum.

The essential character of all tabular aids to computation is, that the results of many operations are recorded in some systematic way for easy reference, and that thereby the computer is spared the toil of obtaining these results for himself.

In many cases this constitutes almost the whole advantage of the table. Thus when, instead of extracting the cube root of some number, we take it from a printed book, we are merely using another's labour. The gain to the calculating community is, that the oft-repeated extraction of the same root is avoided. We also gain by the facility of systematic calculation; the labour of computing a series of successive results being in general only a small fraction of that which would have attended the same work performed in a desultory manner.

It is natural to suppose that soon after it became known that injurious effects follow the introduction of certain substances into the system, attempts were made to remedy these effects, and also to discover counteragents, or antidotes, to the hurtful substances. The success attending these attempts must, of necessity, have been closely related to the existing state of knowledge regarding the actions of active substances. When the effects of poisons were referred to supernatural manifestations, it was chiefly charms and superstitious rites that were trusted to as protectives and remedies. At a somewhat more advanced period in the progress of human knowledge, vague notions of physiological laws and processes supplied the indications of curative treatment.

On the subject of accent and quantity as elements of human speech, there has been such an immense amount of confusion, arising from vague phraseology, that in renewing the discussion nothing seems more necessary than to start with a careful and accurate definition of terms; and that a definition not taken from books, and the dumb bearers of a dead tradition, but from the living facts of nature, and the permanent qualities belonging to articulated breath. Now, if we observe accurately the natural and necessary affections of words in human discourse, considered merely as a succession of compact little wholes of articulated breath, without regard to their signification, we shall find that all the affections of which they are capable amount to four. Either (1), the mass of articulated breath which we call a word, is sent forth in a comparatively small volume, as in the case of a common gun, or it is sent forth in large volume, as in the case of a Lancaster gun; this is a mere affair of bulk, in virtue of which alone it is manifest that any word rolled forth from the lungs of a Stentor must be a different thing from the same mass of sound emitted from a less capacious bellows. In common language this difference is marked by the words loud and low.

There are several problems of great practical importance in electro-magnetic measurements, in which the value of a quantity has to be calculated by taking the sum of the logarithms of the distances of a system of parallel wires from a given point. The calculation is in some respects analogous to that in which we find the potential at a point due to a given system of equal particles, by adding the reciprocals of the distances of the particles from the given point. There is this difference, however, that whereas the reciprocal of a line is completely defined when we know the unit of length, the logarithm of a line has no meaning till we know not only the unit of length, but the modulus of the system logarithms.

Of all the branches of interesting and curious learning, there is none which has been so systematically neglected in this country as mythology—a subject closely connected both with theology and philosophy, and on which those grand intellectual pioneers and architects, the Germans, have expended such a vast amount of profitable and unprofitable labour. The consequence of this neglect has been, that of the few British books we have on the subject, the most noticeable are not free from the dear seduction of favourite ideas, which possess the minds of the writers as by a juggling witchcraft, and prevent them from looking on a rich and various subject with that many-sided sympathy and catholic receptiveness which it requires. In fact, some of our most recent writers on this subject have not advanced a single step, in respect of scientific method, beyond Jacob Bryant, unquestionably the most learned and original speculator on mythology of the last century; but whose great work, nevertheless, can only be compared to a grand chase in the dark, with a few bright flashes of discovery, and happy gleams of suggestion by the way. For these reasons, and to make a necessary protest against some ingenious aberrations of Max Müller, Gladstone, Inman, and Cox in the method of mythological interpretation, I have undertaken to read the present paper; which, if it possess only the negative virtue of warning people to be sober-minded and cautious when entering on a path of so slippery inquiry, cannot be deemed impertinent at the present moment.

The discovery of those numbers which shall, either truly or approximately, represent the ratio of two magnitudes, necessarily attracted the attention of the earliest cultivators of exact science. The definition of the equality of ratios given in Euclid's compilation clearly exposes the nature of the process used in his time. This process consisted in repeating each of the two magnitudes until some multiple of the one agreed perfectly or nearly with a multiple of the other; the numbers of the repetitions, taken in inverse order, represented the ratio. Thus, if the proposed magnitudes were two straight lines, Euclid would have opened two pairs of compasses, one to each distance, and, beginning at some point in an indefinite straight line, he would step the two distances along, bringing up that which lagged behind, until he obtained an exact or a close coincidence.

1. The lunar diurnal variation of magnetic declination as first discovered by Kreil, depended on too few observations to be free from the errors introduced by irregular disturbing causes. The independent discovery of the lunar action on the magnetic needle made afterwards by myself, was liable to the same criticism; but the agreement of the results obtained, both for the magnetic declination and the horizontal force, was sufficiently great to give a considerable value to the conclusion, that the magnetic needle obeys a diurnal law, depending on the moon's hour angle, both as to its direction and the force with which it is directed. This conclusion was farther confirmed in the discussion first made by myself, for the lunar diurnal variation of the vertical magnetic force, which gave, within an hour, the same epochs of maxima and minima as those obtained previously by me for the horizontal component.

In the year 1827, I communicated to the Royal Society of Edinburgh some meteorological journals, kept at Carlisle by the late Mr Pitt, extending over a period of twenty-four years, viz., from 1801 to 1824 inclusive. An abstract of these journals, with explanatory remarks and tabular results, were drawn up by me, and read before the Society, and were afterwards published in their Transactions. I now beg to offer some remarks to the Society on journals kept by Dr Carlyle, in the city of Carlisle, from 1757 to 1783 inclusive, by the Rev. Jos. Golding at Aikbank, near Wigton, Cumberland, fourteen miles west of Carlisle, from 1792 to 1810 inclusive, and by myself at Bunkers Hill, two and a half miles west of Carlisle, which is situate 184 feet above the sea level, according to the late Ordnance Survey, from 1852 to 1870 inclusive. I shall confine my remarks to the quantity of rain that fell during the several periods of our journals. The accompanying tables show the quantity of each month and year included in these periods. I regret much that I am not able to give a description of the instruments used by Dr Carlyle and Mr Golding; but as they both were gentlemen of considerable ability and of liberal education, and devoted much time and attention to meteorology, there is no reason to doubt either the quality of their instruments or the correctness of their observations. Dr Carlyle's rain-gauge was placed in his garden, near the head of Abbey Street, and is about the same height as the ground on which the Cathedral stands, eighty-two feet above the level of the sea. My own rain-gauge consists of a copper funnel, twelve inches in diameter at the top, and is inserted into a strong tinned iron vessel, placed in a box on my garden wall, the height of the funnel being six feet above ground. It is examined from time to time, and particularly after a fall of rain. The water is measured by means of a glass tube of half an inch diameter, with an attached scale of inches and tenths.

In order to determine with exactitude the movements made by the wings in flight, and the part which the air plays in modifying them, I was induced several years ago to collect a large number of facts, and to undertake an extensive series of experiments with natural and artificial wings. My observations and experiments, I may remark, were not wholly confined to flight. On the contrary, I traced the analogy between flying, swimming, and walking; a circumstance which compelled me to pay particular attention to the size, shape, and movements, not only of wings, but also of the travelling surfaces of quadrupeds, amphibia, and fishes. By adopting this method, I obtained suggestions which have proved of the utmost importance to me in my attempts at elucidating the very intricate problem of flight.

I was originally attracted to the study of Quaternions by Sir W. R Hamilton's ingeniously devised and most valuable operator

to which he called special attention (Lectures on Quaternions, § 620) on account of its promise of usefulness in physical applications. But I soon found that in order that its full power may be applied, in general investigations, it is necessary that we should have processes of definite integration, of the kinds required in physics, applicable to quaternion symbols and not merely to scalar variables. I often consulted Hamilton about this want, and he promised to endeavour to supply it at some future time. I fancy that shortly before his death he must have in some way supplied it, though he certainly did not print, nor does he appear even to have written, anything on the subject. In one of the last letters I received from him, he said that he intended to conclude the final chapter of his Elements, which is devoted to physical applications, by some sections on the use of the operator mentioned above. That chapter remains unfinished, and as Hamilton rarely wrote down the steps of even a complex train of mathematical reasoning until he had mentally completed it, it is to be feared that this portion of his investigations is entirely lost. So far as the analytical aspect of Quaternions is concerned, this loss is very serious indeed, for there can be little doubt that Hamilton's solution would have been of immense value from the purely mathematical point of view.

The illustrious Cuvier, in his treatise “ Sur les Ossemens fossiles,” described and figured an imperfect skull which had been obtained, in 1804, by M. Raymond Gorsse in the department of Bouches-du-Rhône, near Fos, on the southern coast of France. It had been found on the sea-shore in the preceding year by a peasant. Cuvier recognised it to belong to an undescribed genus of cetaceans, to which he gave the name of Ziphius; and from the deep hollow which it possessed at the base of the rostrum, he named it Ziphius cavirostris. From the condition of the bones, and the general characters of the specimen, he judged it to be a fossil. Cuvier's description, though brief, and from a mutilated specimen, yet clearly states the most salient features of the skull.

In a paper communicated to the Royal Irish Academy in 1841, I gave an account of a large number of experiments on the heat disengaged when acids and bases, taken in the state of dilute solution, enter into combination, and when bases, insoluble in water, are dissolved in dilute acids. The following general conclusions or laws were deduced from those experiments:—

Law 1.—The heat developed in the union of acids and bases is determined by the base and not by the acid, the same base producing, when combined with an equivalent of different acids, nearly the same quantity of heat; but different bases, different quantities.

Law 2.—When a neutral is converted into an acid salt, by combining with one or more atoms of acid, no change of temperature occurs.

Law 3.—When a neutral is converted into a basic salt, by combining with an additional proportion of base, the combination is accompanied with the evolution of heat.

The Ipecacuan plant, Cephaëlis Ipecacuanha of Achille Richard, has been cultivated in the Edinburgh Botanic Garden for upwards of forty years, but it was not propagated to any extent until 1870, when a proposal was made to attempt the cultivation of the plant in India. This suggestion was made on account of the continued destruction of the plant by the collectors in Brazil, and the risk of scarcity in the supply of this most valuable remedy for dysentery. The Secretary of State for India (His Grace the Duke of Argyll), under the recommendation of several medical officers in Bengal, authorised an attempt to propagate the plant in our Indian possessions, and with that view application was made to me and others to aid in this important undertaking. Accordingly, I at once set about the propagation of the plant in the Edinburgh Garden, with the assistance of Mr M'Nab the curator. He found that the plant could be multiplied very rapidly by dividing the annulated root, cuttings of which, though very small, give off young shoots when placed in favourable circumstances. By this means, numerous plants were produced very rapidly, and the method was also followed by the Messrs Lawson, Nurserymen, Edinburgh, who supplied a large stock of vigorous plants. Mr M'Nab drew up a report of his mode of propagation, which was printed, and distributed extensively to district officers in India and elsewhere. The paper also appeared in the Transactions of the Botanical Society of Edinburgh, vol. x. p. 318.

In the twenty-fourth volume of the Society's Transactions, a very convenient formula is given for computing the time of oscillation in a circular arc; and the investigation of that formula is conducted by an appeal to the actual phenomena. It is defective in so far that it contemplates chiefly the time of oscillation over the whole arc, and does not enable us conveniently to compute the time in which a part of that arc is described.

The object of the present note is to supply that defect, and to present the whole subject in a new aspect remarkable alike for its generality and for its simplicity.