1. A year ago last April, Prof. Tait proposed that I should undertake to derive from Mr Kirkman's polyhedral drawings the alternate ± knots of eleven crossings, thus doing for order 11 what had been done so admirably by himself in orders 8, 9, and 10.

2. The work has been a very protracted one, because of the great number of forms involved—more than three times as many as in all preceding orders combined. Mr Kirkman's manuscript contains 1581 forms, of which 22 are bifilar and 16 duplicates. I find from the remainder 357 knots with 1595 forms as shown in the following table:—

As this is an odd order, perversion doubles these numbers, making 714 elevenfold knots, with crossings alternately over and under.

3. It has been thought unnecessary to show upon the Plates more than one form of each knot; all, however, have been drawn. Knots of each class having the same number of forms are grouped together to make more simple the identification of a particular elevenfold. A small figure following the series number upon the plates indicates how many distinct forms each knot can assume. Knots 84, 357, and 2386 are misplaced.

The rocks which form the Maltese islands have been at various times referred to very different parts of the Cainozoic group. At first they were all assigned to the Eocene, but a further knowledge of their fossils led to the transference of the whole series to the Miocene, in which system they have been included by most English writers. Herr Th. Fuchs, However, who first attempted any precise determination of the horizons of the subdivisions, correlated the Upper Limestone with the Leithakalk and the Blue Clay with the Schlier, and included the two lowest beds in the Oligocene. Dr Murray, however, in the course of his recent Memoir, has explained the striking palæontological differences between the faunas of the different deposits as due to altered conditions of formation rather than to lapse of time, and seems to follow earlier authors in including the whole series in the Miocene.

The Echinoidea of these deposits were originally described by Dr Wright in 1855,§ with additions and corrections in 1864; but since that time much work has been done on the allied faunas.

Some time since I published in the Transactions of this Society a paper upon Phreoryctes; the present paper is the second of what I hope will be a series of memoirs upon the structure of the Oligochæta Limicolæ; this will be a parallel series to that upon the Oligochæta Terricolæ, which is being published by me in the Quarterly Journal of Microscopical Science. As I took pains in my paper upon Phreoryctes to point out, in accordance with views previously expressed by myself and by others, that it is impossible to draw a hard and fast line between those two groups of Claparède, it may seem rather illogical to retain this classification—if only in the title of a paper; I do so simply as a matter of convenience, and without in the least desiring to revive these old divisions of the Oligochæta; indeed the genus Moniligaster, which is treated of in the present communication, is by most naturalists regarded as an earthworm; in many points it does undoubtedly agree with certain terricolous genera; but as its affinities, into the discussion of which I shall enter later, seem to me to be more with the Lumbriculidæ, I put it into the Limicoline series; it is useful to have a name corresponding to “earthworm” for those Oligochæta which are not earthworms, and are for the most part aquatic, and I therefore use that term.

[A few words are necessary to explain why the present paper has hitherto been printed in Abstract only, and to show what modifications it has undergone since it was read more than two years ago.

In the paper, as it was first presented to the Society, I contented myself with the usual practice of extracting from the virial a negative term (— βp) to represent at least a portion of the part due to the molecular repulsion at impact. But, as will be seen by the Abstract printed at the time (Proc. Roy. Soc. Edin., 21/1/89), I stated that though this procedure is correct when molecular attraction is not taken into account, it requires considerable modification when such attractions are introduced. I also stated that its main effect would be to alter one of the disposable quantities (A) in my equation. I have since seen that the definition, of what we are now to understand by “temperature,” which I then introduced, leads naturally and directly to the writing of a part of A in the form

−e(E+C/(v+γ)),

where E is proportional to the absolute temperature and to the average energy of a free particle. This remark really substitutes the new undetermined quantity e for the β which occurred in my former expression.

The development of the inverse square root

into a series

gives rise to the coefficients Zn, which have been called “Laplace's Coefficients” If in Zn we substitute for z the expression

where

then the function of ψ, which Zn will represent, can most appropriately be expressed by

The object of this paper is to show, by actual calculation, that the coefficients Ns, (functions of x, x′, y, y′) can be worked out by elementary algebraical processes, the only auxiliary taken from higher analysis being the expression of the powers of cos ψ in function of the cosines of the multiples of ψ.

As to our notations, we shall observe throughout the following restrictions:—

1. The function Π(x) shall not be employed otherwise than for integer positive values of x, so that

2. The factorial

will be made use of for integer values only of r (as, for example, r = + 1, or r = − 1, or r = + 2, &c., the exponent t being integer positive.

At present our knowledge of the exotic genera of the aquatic Oligochæta is not very far advanced. During the last twenty years there has been a considerable accumulation of descriptions of exotic Earthworms, but the lower Oligochæta have been much less studied. The principal investigations into this group have been carried on by Eisen, who has made us acquainted with a number of interesting forms, belonging to the families Tubificidæ and Lumbriculidæ, from North America. Other naturalists, such as Leidy, have also dealt with the Oligochætous fauna of that country; but their papers have chiefly had for their object the discrimination of genera and species, and are not so much concerned with the description and delineation of anatomical structure. Beyond the series of papers published by the above-mentioned authors, we have only a few scattered memoirs by other writers upon exotic species of “Limicolous” Oligochæta.

Having recently been awarded, by the Government Grant Committee of the Royal Society, a sum of money to assist me in the investigation of the Oligochæta, I have been anxious not to limit myself to Earthworms, but to obtain as many specimens of the aquatic forms as possible.

The fjord-like inlets or sea-lochs which form so conspicuous a feature in the scenery of the west of Scotland stand in marked contrast to the shallow, low-shored firths of the east coast. When Dr John Murray decided to extend the physical and biological work of the Scottish Marine Station to the west coast he foresaw that many interesting conclusions were likely to be derived from the study of these isolated sea-basins. Various papers, published by him and other workers, contain preliminary discussions of many of the phenomena observed, fully justifying the anticipations which had been formed.

For one year my work, as described in this paper, was carried out under the provisions of an Elective Fellowship in Experimental Physics of the University of Edinburgh, to which I had been elected in 1886; and subsequently by a personal grant from the Government Grant Committee for Scientific Research. The Committee also devoted several sums of money in payment of expenses in compiling this discussion. The Scottish Marine Station throughout gave the use of the steam-yacht “Medusa,” and the necessary apparatus.

I will commence by stating that three reasons have moved me to bring this subject before the Society—(1) Because I found everywhere loose and even altogether false ideas possessing the public mind on the subject; (2) because I much fear that we, the academical teachers of the Greek language, are chiefly to blame for the currency of these false ideas; and (3) because, if Greek is a living and uncorrupted language, and dominating large districts of Europe and the Mediterranean, as influentially as French on the banks of the Seine and German on the Rhine, it follows that a radical reform must take place in our received methods of teaching this noble and most useful language. Now that the current language of the Greeks in Athens and elsewhere is not, in any sense, a new or a corrupt language, as Italian is a melodious and French a glittering corruption of Latin, may be gathered even a priori; for languages are slow to die, and the time that elapsed from the taking of Constantinople by the Turks in 1453 and the establishment of the Venetian power in the Morea in 1204, to the resurrection of Greek political life in 1822, was not long enough to cause such a fusion of contrary elements as produced the English language from the permanent occupation of the British Isles by the Normans.

From a given square one quarter is cut off; to divide the remaining gnomon into four such parts that they shall be capable of forming a square.

[See Trans. Roy. Soc. Edin., xxi. 271.]

Preliminary problem, viz.: To cut a rectangle into three parts capable of forming a square (Figs. 1, 2).

Lay off AX and CY = side of square, join BX, draw YZ ∥ AB, and 1, 2 and 3 will be the required pieces, as is evident.

The problem will be impossible when the length of the rectangle is greater than four times its breadth; four or more pieces may then be required. The problem can be solved in four different ways by drawing the sloping line BX from each of the four angles.

The above is a particular case of the more general problem, viz., to cut a rectangle or a parallelogram into three pieces so as to form another with a given side, as AB (Figs. 3, 4).

In the gnomon (Figs. 5,6), suppose the square EC placed below in the position BK, then the rectangle AK is formed, and can be divided into three pieces by FX and YZ, so as to form a square as before.

The appearance of a learned and exhaustive work on the life and labours of Koraes, by a native Greek of great ability, naturally invites the scholars of Western Europe to a grateful acknowledgment of the obligation of the Greek language to this most distinguished of its modern exponents. The fact, indeed, that Greek in this country is popularly talked of as a dead language, and as such taught from books through the eye and the understanding, rather than by living converse with those who speak it, may serve as an apology for the general ignorance that prevails, even in professional circles, with regard to the scholarly achievements of this remarkable man; but the appearance of works of such mark in the living literature of Greece as those of Thereianos, Paspates, Bikellas, Polylas, and others, warns us that it is high time to take note of living Greece as living Greece again, and give to the learned Grecians of the present day their proper share in that homage which we pay so liberally to the great masters of Greek wisdom in the past.

Adamantios Koraes was born at Smyrna in 1748, the son of a Chiote merchant who had removed from the island to the great centre of commerce on the Continent. As a boy young Koraes was remarkable for his love of learning and his hatred of the Turks as the enslavers and debasers of his people; and having inherited a valuable library from one of his maternal relatives, he found it necessary to acquire the Latin language, in which all the famous commentaries on the great Greek classics had been composed.

Towards the end of May of this year I made my third visit to the Rigi Kulm, for the purpose of continuing my observations on the amount of dust in the atmosphere and other meteorological phenomena. It was with a feeling of no little satisfaction that I found myself at that elevated situation during broken weather, and under conditions very different from any previously experienced by me. On this occasion I had come in the hope of finding opportunities for making some observations on the conditions existing in clouds, in addition to the usual dust observations, and had brought with me the small instrument for observing the water particles in a fog, described in a previous communication to this Society. As the weather continued to be variable during the week of my visit, I was fortunate enough to succeed in making a number of interesting observations on the water particles in clouds, and also of comparing the conditions in clouds at this elevated situation with those previously observed in fogs at a low level.

Before giving the results of my observations on the water particles, it may be desirable to make a few remarks on the solid or dust particles. When making the ordinary dust observations this year, it was frequently noticed that, when surrounded by clouds, the number of particles varied greatly at short intervals of time.

The present paper is the second of the series dealing with the Fossil Flora of the Staffordshire Coal Fields. As in previous memoirs, I give a short sketch of the Geology of the coal field, merely for the purpose of indicating the relationship of the beds to each other, from which the fossils have been derived.

Various memoirs dealing with the geological structure and resources of the Potteries Coal Field have already appeared, but in these the names applied to the different groups of strata which compose the Potteries Coal Field have generally special application to the local geological features, and do not treat of the Coal Field in its wider relationship, when considered as only forming a part of the Coal Measures as developed in Britain. A similar course has usually been taken in the published memoirs of other British Coal Fields, which makes a comparison of their relative ages, from the data given, very difficult.

Although the Mollusea have usually been collected and examined, from their great vertical distribution—in some cases extending throughout the whole range of carboniferous rocks—they as a whole afford little data for the determination of the divisions of the Coal Measures, and unfortunately the fossil plants appear to have received little attention when the memoirs of the various coal fields were being prepared.

Clinical and pathological observation both show that the dentine of the tooth is very closely connected with the nervous system, and is in consequence highly sensitive. Upon what structures does the sensibility of the dentine depend? In what manner is the dentine connected with the nerves of the pulp so as to become so sensitive to external stimuli?

Perhaps there is no other structure in the body which is so largely supplied with nerves as the pulp of the tooth; even in the smallest fragment we find many nerve fibres. If we take the pulp from the incisor tooth of an ox and examine it after having allowed it to lie in a solution of osmic acid for a few minutes, we can see clearly through the darkened semi-transparent tissue a large blackened nerve trunk passing up the centre of the pulp, giving off on its way innumerable lateral branches, and dividing in a brush-like manner near the upper part of the pulp. All the fine branches are directed towards the periphery of the pulp. In longitudinal sections of the pulp we can see the same in greater detail; many large bundles of medullated and non-medullated nerve fibres running longitudinally near the centre and giving off lateral branches, which are found in great numbers near the periphery and divide into single nerve fibres just under the odontoblastic layer, being specially numerous at the apex of the pulp.

At the end of 1886 a method occurred to me of rapidly recording the positions of the lines of the solar spectrum, which I thought might be used with advantage for determining the faint “Telluric dry-gas lines” near D, mentioned in Professor Piazzi Smyth's maps of The Visual Solar Spectrum in 1884. The fundamental idea of the recording apparatus is that of magnifying by some mechanical means the motion of the grating, or prism, to such an extent that it can be recorded on a continuous fillet of paper. The viewing telescope being then firmly clamped, the exact positions of the grating can be pricked off on the strip of paper as the lines are successively brought to the fixed cross in the field of view.

For the satisfactory use of the method two conditions suggest themselves as desirable. First, the punctures should not be less than a good-sized pin hole, and the interval between the closest lines, which the spectroscope is able to separate, should be represented on the paper strip by a space of several tenths of an inch.

All observers are agreed that the bones of the plastron and some bones of the carapace are simple membranous bones, arising from centres formed in a pre-existing fibrous membrane. Considerable difference of opinion exists as to the development of the costal and neural plates of the carapace.

Owen speaks of the ossification “as extending from the ribs and neural spines into the substance of the neural and costal plates. The ribs and spines enter into the composition of the carapace.”

Gegenbaur questions whether the ribs of the Chelonia “are not in reality enormously developed transverse processes, and considers that the neural and costal plates have developed in the integument.”

Claus remarks that the spinous processes of eight of “the thoracic vertebræ (2nd to 9th) appear in the middle line as horizontal plates (neural plates), the ribs of the same vertebræ are transformed into broad transverse plates (costal plates).”

Huxley says that the neural plates and the costal plates exist as expansions of the cartilages of the neural spines and ribs of the primitive vertebræ, before ossification takes place. This being the case, the “neural and costal are vertebral and not dermal elements, however similar they may be to the nucleal, pygal, and marginal plates.”

Forty years ago Kolbe showed that a strong aqueous solution of potassium acetate, when subjected to the influence of the electric current, is decomposed with formation of the following products. At the anode a gaseous mixture is evolved which consists chiefly of carbonic acid and ethan; while at the cathode hydrogen escapes, and potassium hydrate is formed in the solution. When dilute solutions of the same substance are electrolysed under similar conditions, the decomposition products at the cathode are the same as in the previous case, but at the anode the gas evolved is now oxygen, and free acetic acid makes its appearance in the solution. The difference between the two cases must, on modern views of electrolysis, be attributed to the occurrence of secondary reactions. The primary process is in all cases the transference of the electrically-charged submolecules or ions to the corresponding poles, where they lose their charges, and thereby become capable of reacting with one another or with neighbouring molecules. As a general rule they are, when discharged, themselves incapable of independent existence.