The extraordinary processes developed in connection with the base of the suspensor in the common Indian Cress (Tropæolum majus) have long excited the interest of vegetable embryologists, and have led to the publication of numerous observations on the embryogeny of that plant. As the following remarks are interesting chiefly in regard to the peculiar features in which the germs in two other species of the genus differ from that in the common Indian Cress, it will not be out of place for me briefly to recapitulate the principal facts connected with the development in that plant.

Proposition I.—All spoken language is a growth, subject, like the creature who uses it, to a constant course of mutation; it is a living organism, developed according to certain laws, partly inherent, partly superinduced; and, though it is liable to decay, disintegration, and death, this disintegration, except in special cases of extermination, becomes the soil of a new growth, and this death the cradle of a new life. The historical action of this process of mutation is to produce either new varieties or dialects of one language, or new species of one family of languages.

I have to explain at the outset that the title of my paper has been chosen for the sake of brevity. Both the words “expiatory” and “substitutionary” are liable to be misunderstood and abused in argument. I shall, therefore, avoid the word “expiate” as much as possible; but as I cannot do without a frequent use of the words “substitution” and “substitutionary,” I merely note at present that there are various modes of substitution and various purposes served by it, and that it is very important in an argument to state accurately both the mode and the purpose.

The object of my paper is to place the two prevalent theories of the origin of sacrifice in juxtaposition with the facts which we have in regard to Greek sacrifices, in order that we may see what light the facts may throw on the theories. For this purpose I first state the two theories.

The extension which is required to be given to the meaning of algebraic addition and subtraction, for the purpose of representing a vector by a polynomial expression (by the algebraic sum, namely, of components differing in direction from one another), renders it necessary to investigate into the question of what the geometrical signification of the result of the other elementary operations of algebra will become when these operations are performed on those polynomial expressions?

In Crelle's Journal, vol. x., Stern devotes fifteen pages (pp. 245-259) of his Theorie der Kettenbrüche to the examination and elucidation of the following problems:—

as continued fractions of the form

where a1, a2, a3, … are independent of x.

In February 1831 the late Mr Witham read to the Royal Society of Edin burgh, a paper of much interest on two fossil trees of great size which had been brought to light, the one in 1826, and the other in 1830, during the excavations carried on in Craigleith sandstone quarry, in the immediate neighbourhood of our city. In March of the same year this paper, with the addition of several chemical analyses of the fossils, was read also before the Natural History Society of Northumberland. In 1833 he included his observations on these fossils in a separate and more comprehensive treatise on “The Internal Structure of Fossil Vegetables found in the Carboniferous and Oolitic Deposits of Great Britain.” The main purpose of Mr Witham's researches—in addition to an accurate description and delineation of natural objects previously little investigated—was to show that fossil vegetables in the oolitic and carboniferous formations were not, as had been generally supposed prior to his researches, always ferns, tree-ferns, lycopodiums, and other acrogenous plants, but on the contrary many of them woody exogenous trees, showing longitudinal ligneous bundles, transverse medullary rays, concentric annual layers, and other characters belonging to the intimate organisation of our existing forest trees. Mr Witham even went so far as to identify the structure of the Craigleith fossils with that of our modern pines, and to assign them to two separate species.

Sir Charles Lyell, in the 3d edition of his “ Antiquity of Man,” published in 1863, makes the following statement:—

“Until lately, it was confidently assumed, that no alteration had. occurred in the relative levels of land and sea, in the central district of Scotland, since the construction of the Roman or Pictish wall (the wall of Antonine), which reached from the Firth of Forth to that of the Clyde.”

The mode of growth and increase amongst the Coralligenous Actinozoa in general has been more or less fully treated of by various observers, including such distinguished naturalists as Milne-Edwards, and Haime, Dana, Martin Duncan, Fromentel, Agassiz, and others. I do not, therefore, in the present communication propose to pass the whole of this subject in review, but rather to consider the general and special peculiarities of growth and non-sexual reproduction exhibited by the corals of the Palæozoic Period alone. Many of these peculiarities are of great interest, both from the stand-point of the palæontologist, and also as concerns the systematic zoologist, and they have not yet met with all the attention they deserve. To carry out this inquiry, it will be necessary first to consider the general phenomena exhibited by the Palæozoic corals, as regards their mode of growth and increase. We may then examine the bearing of these phenomena upon various points connected with the classification of these ancient corals, and more especially upon their generic and specific affinities and differences. Finally, we may briefly consider the relations which exist between different parts of a compound corallum as regards their growth, and their influence upon the ultimate form of the colony.

The proposed scalars represent the differentials of the three functions, X, Y, Z, of the co-ordinates x, y, z, put under the form

where ◁ represents the operator

and where

.

These functions, equated respectively to constants, are intended to represent a system of orthogonal surfaces.

In preparing solutions of various salts with a view to determine their electrical conductivity, we found it of importance to know both the amount of salt in each solution, and its density. We prepared the solutions by mixing known weights of the salt under examination with known volumes of distilled water, and then measured their densities as soon as solution was complete by weighing a glass bulb in the liquid. We employed a balance which weighed to a milligramme with a kilogramme in each pan, and to give the results as great accuracy as possible we corrected for the upward pressure of the air displaced by the weights. The weight of the bulb itself was about eighty grammes. When not in use it was kept immersed in distilled water, in order to prevent change in its weight or volume being caused by impurities adhering to it. The temperature at which the densities were taken was 10° centigrade.

§ 1. The general problem in elasticity, as usually presented for solution, supposes the elastic substance to pass initially from a state without strain. But important cases exist where it must be conceived to start from a state already under considerable stress. When this is the case, the constitution of the solid undergoes great change, as is shown by the fact that strained glass loses its isotropic property, and becomes doubly refractive. This subject was long ago attacked by Cauchy, who, by means of the theory of molecular actions, deduced the existence, in the expressions for the stresses due to the secondary strains, of terms proportional to the initial or primary stresses. The problem has been since discussed by MM. De St Venant and Boussinesq, who have applied to it Green's expression for the energy stored up during the strain. But the question, in their hands, still retains traces of Cauchy's hypothetical element, inasmuch as their expression for the potential energy was deduced by means of the molecular theory. M. De St Venant even considers it a strong argument for the truth of the latter, that it is indispensable in the discussion of this problem. These authors have also failed to see in what way the remaining part of the potential depends on the original strain.

In the series of chapters of which the present is the first, I purpose to submit the results of an analytical examination of all the minerals of Scotland whose composition appeared doubtful; of such as had not previously been examined; or of such as appeared in any way to be of special geologic interest.

Comparative Anatomists have long been desirous to obtain detailed information on the Placentation of the Sloths. The only observations on this subject which appear to have been made up to this time have been recorded by Professor Rudolphi of Berlin, and Dr C. G. Carus of Dresden. Rudolphi in the course of some remarks on the structure of the umbilical cord in Bradypus tridactylus incidentally mentions that the placenta was cotyledonary as in the Ruminants. Carus figures the placenta and an almost mature fœtus of a B. tridactylus which came into his possession in 1830. He gives no description, however, of the specimen, but contents himself with a brief explanation of his engraved figures; in the course of which he says, that the specimen seemed to him important on account of the length of the umbilical cord, and the form of the cotyledons, which did not project, as is usual, outwards, but towards the inner face of the ovum, a peculiarity which had not yet been observed and described.

Observations on the form of the Placenta and on the arrangement of the Fœtal membranes in the Seals have been recorded by more than one anatomist. Alessandrini of Bologna obtained a specimen of the Monk Seal (Phoca bicolor) Monachus albiventer, in the gravid condition, in which the uterus contained a single fœtus.

The method of solution which I have the honour to lay before the Society is founded upon an old and well-known arithmetical process called “ casting out the nines.” It consists in substituting for any number the remainder which that number leaves when divided by nine. As this remainder contains only a single figure, whereas the original number may contain any number of figures, it is obvious that if the remainder can be employed instead of the number itself, a great saving of time and trouble must follow.

A few years ago, a memoir on high-water marks on the banks of the Rivers Earn and Teith, in Perthshire, by the Rev. Thomas Brown, was read in our Society, and published in our Transactions.

The only other Scotch geologist, so far as I know, who has alluded to the existence of river terraces, much above the level of existing floods, is the late Dr Robert Chambers. In his work, entitled “Ancient Sea Margins,” Dr Chambers specifies many Scotch rivers, in the valleys occupied by which, he had seen terraces, at considerable heights above the rivers and above the sea.

The following pages contain, in a comparatively compact form, part of the substance of a voluminous paper read to the Society six years ago. Of that paper, which employed ordinary analysis alone, only a few pages had been put in type when I succeeded in overcoming a formidable difficulty which had presented itself in my quaternion treatment of the subject. I therefore withdrew the paper in order that it might have the benefit of the simplification which quaternions always give; but it is only of late that I have found time to complete part of the translation into the new language. From the circumstances under which the paper has thus been produced, i, j , k come forward with undue prominence, a thing to be regarded (in Hamilton's words) “ as an inelegance and imperfection in this calculus, or rather in the state to which it has hitherto been unfolded.” Immense as is the simplification already attained, it is clear that in many places still more is attainable. But I have not postponed my paper till it should receive this final polish, partly because the time I can devote to such inquiries is extremely limited, and partly because I think that several of the results obtained, and of the modes of obtaining them, are new and remarkable. Besides, a question of this order of difficulty is admirably adapted to show in what respects quaternion methods require improvement. There must be some simple mode of deducing (13) and (21) below from (7) without the explicit use of i, j , k, but I have not yet been fortunate enough to discover it.