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.

[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.

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.

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.

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.

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.

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.”

In former papers on the pharmacological action of Strophanthus, dating from 1869, I selected for description, from the considerable number of experiments that had been made, merely those experiments which sufficed to illustrate the general features of the action, and especially such effects as seemed likely to form a basis for the application of Strophanthus to the treatment of disease.

I had intended to have followed, at no distant date, these preliminary and somewhat fragmentary notices by a more complete description of the pharmacological action, for which, indeed, nearly all the required experimental data had several years ago been obtained; but unavoidable circumstances prevented this intention from being fulfilled. In this part of the present paper the fuller description will be given; and if any excuse were required for doing so, it may perhaps be found in the circumstance that the anticipation of the therapeutic value of Strophanthus has been amply confirmed by the important position now occupied by it as a therapeutic agent.

The following analyses were made some years ago, principally with the object of ascertaining the state of oxidation of the manganese in the nodules. The nodules examined came from three different localities, two of them oceanic and the third littoral. Samples marked I., II., and III. are from nodules brought up in the trawl on board the “Challenger,” on 13th March 1874, in lat. 42° 42′ S., long. 134° 10′ E. The depth of the water was 2600 fathoms, and the temperature of the bottom water 0·2° C. The density of the bottom water was 1·02570 at 15·56° C. Being from a high southern latitude, and therefore near the source of surface aeration, the water is highly charged with atmospheric gases, especially oxygen. It contained, per litre, 18·4 c.c. of mixed nitrogen and oxygen, of which 31·81 per cent, was oxygen, and 27·33 c.c, or 53·7 milligrammes, loosely-bound carbonic acid. The position of the station is about 400 miles south-west of the nearest part of the Australian coast, and about 500 miles west of Tasmania. It was the deepest water observed in the Antarctic voyage between the Cape of Good Hope and Melbourne. The haul was a very abundant one, and a few notes which I made at the time may be interesting:—“The water was found unexpectedly deep, the bottom being red clay, with some Foraminifera.

This paper forms the continuation of a paper already communicated to the Royal Society of Edinburgh, and published in the Transactions (vol. xxxv. pp. 377–390). Strictly speaking, it is third of a series bearing on the same subject. In the first I showed that when a current is passed along a longitudinally magnetised nickel wire a twist is produced similar in character but opposite in direction to the twist which Wiedemann discovered to be produced in iron similarly treated. This was the chief result arrived at; but other results were also obtained, especially with regard to the influence of tension, which called for further investigation. In the later paper, the influence of tension was more thoroughly studied; and also the effect of change of temperature. I also discussed more fully the suggestion first made by Maxwell that the effect discovered by Wiedemann can be explained in terms of the changes of length which Joule observed in iron when magnetised. Obviously the same explanation must be extended to the case of nickel, which Barrett had found also to be subject to changes of length when magnetised. Mr Bidwell's recent elaborate measurements of these changes of length enabled me to make an approximate calculation of the twist in an iron tube longitudinally and circularly magnetised. The comparison of the result as calculated for a tube with the result observed for a wire of the same diameter, established, in my opinion, the sufficiency of Maxwell's explanation.

Noting the direction of the wind is part of the regular routine of observations at the Observatory on the summit of Ben Nevis. Owing to the climatic conditions, especially the frequency of frozen fogs in winter, anemometers are practically useless, but the record is made complete enough for many purposes by the system of hourly observations; at each hour of the day and night the wind, at the time of making the observation, is noted, and the number of entries of each wind direction in the day, month, or year must very nearly represent the relative frequency of such winds for that time. Table I. is a summary giving the number of times each wind is observed in each month of the year on an average of six years (1884–89); February being given in duplicate, that in the body of the table containing 28 days, and that at the foot 29, with the winds divided in the same proportion. The wind is observed to 32 points, but the entries have been reduced to 8 points for this paper by counting all the bys to their adjacent octant, and halving the 8 intermediate points between the octants; thus all N. by E. count as N., all N.E. by N. as N.E., while the N.N.E. entries are halved between N. and N.E., and so on. Perhaps a stricter method of reduction would have been to divide the intermediate points in the ratio of the relative frequency of the adjacent octants, but this would have been considerably more laborious, and of doubtful utility; still it should be borne in mind that the method used tends to smooth out any sudden changes in the frequency of winds of nearly the same direction.

It seems strange that a principle so fundamental and so widely used as Lagrange's Rule for Solving a Linear Differential Equation should hitherto have been almost invariably provided with an inadequate demonstration. I noticed several years ago that the demonstrations in our current English text-books were apparently insufficient; but, as the method by which I treated Linear Partial Differential Equations in my lectures did not involve the use of them, it did not occur to me to analyse them closely with a view to discovering in what the exact nature of the defect consisted. The consideration of certain special cases recently led me to examine the matter more closely, and I was greatly surprised to find that most of the general demonstrations given are vitiated by a very obvious fallacy, and in point of fact do not fit the actual facts disclosed by the examination of particular cases at all.

The worms which form the subject of the present communication were forwarded to me, in a living condition, from Kew Gardens.

I have received lately a considerable number of living Oligochæta from those Gardens, through the kindness of Mr Dyer, who permitted me to have the earth arriving from different parts of the world in the Wardian cases, in which plants are packed for travelling, thoroughly sifted, with a view to preserving the worms which had been accidentally included. By these means I have succeeded in obtaining some very interesting new forms, as well as a number of others which are still imperfectly known. The species which I describe in the present paper appears to be a new species of Eisen's genus Ocnerodrilus. The genus Ocnerodrilus was formed by Eisen in 1878 [1] for a small worm found in Fresno County, California. The specimens were all met with in “an irrigation box,” where they were found crawling among the algæ which covered the boards. It is evidently, therefore, aquatic in its habits, but Eisen contrasts its slow movements with the rapid swimming of Lumbriculus and Rhynchelmis, comparing it in general appearance with a small species of Lumbricus.