The new species here described were found in the course of the work of the Lake Survey on Loch Ness and other Highland Lochs. Half of the number were found in lakes, though they are not exclusively lacustrine, three in ponds, two from moss growing on the shores of Loch Ness, and one in a stream running into Loch Ness.

The following paper deals with an investigation of the successive zones of plant remains contained in the deeper peat deposits covering areas in the Scottish Southern Uplands. The field work was carried on during the summer and early autumn of 1904, and the detailed examination of the peat in the laboratory during part of the winter. No attempt has been made to work out the detailed flora of the different zones, but attention has chiefly been directed to the dominant plant remains found at different horizons in the mosses. Whilst the list of plants from each zone is small, the general facies of the flora of any layer can be gauged from the abundant presence of a few characteristic plants such as Salix reticulata and Empetrum, or Sphagnum and Eriophorum. Thus, while the investigation is incomplete as regards any addition to the history of the British Flora, it will, I hope, throw some light upon the succession of vegetation over the older peat mosses since their origin.

(41) The invariance of the equations

with regard to the group of cyclical substitutions,

and the consequent invariance of the eliminant with regard to the reduced group consisting of the last two substitutions, has been already referred to. When the eliminant is expressed in terms of the three-line determinants formable from the array of coefficients, the invariance in question is self-evident, as each of the twenty-eight parts composing the expression is invariant by itself. For convenience this form may be repeated from § 31 with a slightly improved notation.

In this paper, theorems which are extensions of the following, are discussed:—

We define J[n] (λ, x) as

In this expression

The theory of the functions commonly known as q functions might perhaps be greatly developed, if investigators were to work on lines suggested by the functional notation of well-known analytie functions. For instance, the analysis connected with the circular functions sin x, cos x, …, might be regarded as the theory of certain infinite products without using any special functional notation. It need not be explained however, how great was the gain to elementary algebra by the introduction of the exponential function (regarded as the limit of a certain infinite product, or as the limit of a certain infinite series) denoted ex, with certain characteristic properties, enabling the worker to make transformations easily and quickly. Of course, the vast store of interesting and in many cases useful results connected with the elementary functions of analysis might have been obtained without the introduction of any notation capable of rapid and easy transformations, but I think it unlikely that they would have been obtained.

§ 1. The networks considered in the following paper are those networks of the plane whose meshes are regular polygons with the same length of side.

When the polygons are all of the same kind the network is called regular, otherwise it is semi-regular.

The regular networks have been investigated for the three geometries from various standpoints, the chief of which may be noted.

1. The three geometries can be treated separately. For Euclidean geometry we have then to find what regular polygons will exactly fill up the space round a point. For elliptic geometry we have to find the regular divisions of the sphere, or, what is the same thing, the regular polyhedra in ordinary space. The regular networks which do not belong to either of these classes are then those of the hyperbolic plane.

My main purpose in the following short paper is to publish figures illustrating some of the more important facts of the early development of myotomes and motor nerves in Lepidosiren. The bearing of some of the observations of nerve development upon current theories renders it particularly desirable that they should be illustrated by untouched photographs of the sections. A few photographs illustrating the more important stages in the development of the motor nerve trunks are given on Plate I. For the preparation of the photographs here published, as well as several others, I am indebted to the skill of my friend Dr T. H. Bryce, and of Mr Fingland, our University photographer.

The following paper contains a purely analytical discussion of the problem of the deformation of an isotropic elastic plate under given forces. The problem is an unusually interesting one. It was the first to be attacked (by Lamé and Clapeyron in 1828) after the establishment of the general equations by Navier. The solution of the problem of normal traction given by these authors, when reduced to its simplest form, involves double integrals of simple harmonic functions of the coordinates. The integrals are of complicated form, and practically impossible to interpret, a fact which, without doubt, has had much to do with the neglect of the problem in later times, and the almost complete absence of attempts to establish the approximate theory on the basis of an exact solution. An even more serious defect of Lamé and Clapeyron's solution is that the integrals, as they stand, do not converge. A flaw of this sort has often been treated lightly by physical writers, the non-convergence of an integral being regarded as due to the inclusion of an infinite but unimportant constant. In the present case, however, the infinite terms are not constant, but functions of the coordinates, and the modifications necessary to secure convergence, so far from being unimportant, lead directly to the most significant terms of the solution.

In September 1903 I received a bottle containing twelve specimens of a large parasite presented to me by Mr Chr. Castberg, the manager of a Norwegian whaling company which has established a fishing station at Ronasvoe in the north of Shetland. In his letters Mr Castberg stated that the parasites were attached to a Finner whale, which, from its size, the mottled character of the whalebone and the pointed head, was obviously a Razorback—Balænoptera musculus. The parasites were numerous, and were fixed to the back of the whale, and the attached end penetrated through the skin into the blubber. Although Mr Castberg had seen many hundred whales, this is the first occasion on which he had met with this form of parasite.

The present work was commenced in the summer of 1902, with the object of writing a complete monograph of the morphology of Myxine. It had previously been undertaken by the late Professor G. B. Howes, F.R.S., but, owing to the pressure of other work, and the first indications of the illness which subsequently proved fatal, he handed the work over to me. During the earlier stages, however, he exhibited a characteristic interest in the research, and most generously placed his material and the late Dr Pollard's sections at my disposal. His death has removed a kindly and a stimulating figure from our midst; a man whose life was devoted to the service of his friends and the advancement of his favourite study, and whose generous and sympathetic nature was the wonder and the delight of all who knew him.

The present communication is intended to be the first of a series dealing with observations on the life-history of the Anura Aglossa and their anatomy at different stages of development. Xenopus lævis, with its small ova and protracted larval free-swimming stages, must necessarily form a basis for the study of the development of that other remarkable Aglossan, Pipa americana. Although the adult Aglossan is an aberrant and specialised Anuran, there are Urodele features in the development of Xenopas which make its embryology of great general interest. These primitive features, combined with others peculiar to the genus, impress a character upon the early life-history of this frog which is widely divergent from that of the Phaneroglossa with small ova.

In the first part of this memoir I described the structure of the corpuscles at a stage of larval development when the red cells were actively dividing and the blood contained several varieties of white cells. During the course of these more strictly cytological observations, it was impressed upon me that the great size of the elements and their very marked histological characters, combined with the simple character of the organisation of the animal, made Lepidosiren a very favourable case for the study of the first principles of Hæmatogenesis. I was specially interested in what may be termed a middle phase in the history of the blood. I refer to a period after the primitive corpuscles have acquired haemoglobin and there are leucocytes present, but before the blood-forming organs are unfolded. This stage lasts a relatively long time in Lepidosiren up to the differentiation of the spleen, as the liver takes no part in blood-formation at any period.

§ 1. In a paper published in the present volume of the Transactions of the Society (p. 659, 3rd July 1905), one of the authors of this communication has given a Hydrodynamical Theory of Seiches, and deduced formulae with a view to particular applications. The object of the present paper is to give some account of a series of calculations on which we have been occupied for some time, with a view to work out, a priori, from the Bathymetric Data of the Lake Survey, the Seiche Constants for the three seiches of lowest nodality in Lochs Earn and Treig.

§ 2. As this is the first time that any attempt has been made to solve completely a problem of the kind, we naturally selected lakes having as simple a configuration as possible. A general idea of the shape of the two lakes, and also of the nature of the data from which we have worked, will be obtained from the 3-in. maps reproduced in figures (3) and (4).

Since the publication, about a year ago, in the Transactions of this Society, of my paper on the Lower Devonian Fishes of Gemünden, a review of it has appeared in the pages of Science by Professor Bashford Dean, of New York. In this review Professor Dean endeavoured to throw doubts on the correctness of my orientation of the elements of the dermal skeleton of Drepanaspis Gemündenensis, in the following words:—

“Thus, his grounds seem inadequate for distinguishing dorsal and ventral sides. In no specimen figured is the relation of the dorsal lobe of the tail shown convincingly to be continuous with the so-called dorsal aspect; moreover, the eyes occur on the side which Traquair regards as ventral. Unless additional evidence is forthcoming, it would accordingly seem to me more probable that the ‘labial’ of Traquair was the ‘rostral’ plate, a structure which appears constant in Heterostracans.

The Aleyonarians collected by Mr W. S. Bruce on the Scotia voyage represent nine species—six of which are new, namely:—

Primnoisis ramosa, n. sp.

Thonarella brucci, n. sp.

Amphilaphis regularis, Wright and Studer.

Primnoella scotiæ, n. sp.

Primnoella magellanica, Studer.

Paramuricea robusta, n. sp.

Gorgonia wrighti, n. sp.

Gorgonia studeri, n. sp.

Umbellula durissima, Kölliker.

Apart from the six new species, the collection is of interest in extending our knowledge of the geographical distribution of previously recorded forms. Thus Amphilaphis regularis, Wright and Studer, previously collected off Inaccessible Island, Tristan da Cunha, and off Nightingale Island, was got in abundance off St Helena; Primnoella magellanica, Studer, previously collected off Monte Video and in the Magellan Straits, was obtained at Burdwood Bank 54° 25′ S., 57° 32′ W.; while Umbellula durissima, Wright and Studer, previously obtained by the Challenger from the North Pacific Ocean, south of Yeddo, was found by the Scotia at 48° 06′ S., 10° 5′ W.