Since the publication in 1840 of a brief but valuable memoir by J. D. Forbes, in which that author drew attention to “the traces of ancient glaciers” in the Cuillin Hills, that district has remained almost unnoticed by glacial geologists for half a century. This neglect is doubtless attributable chiefly to the difficulty of access to the mountains, a consequence of their peculiar configuration, which in turn is closely bound up with the glacial history of the district. The present contribution is the outcome of observations made during the years 1895–1900 in mapping the central part of Skye for the Geological Survey of Scotland. In traversing the mountains day after day throughout several successive seasons, the writer has been struck especially by the impressive evidence which they present of glacial erosion as the dominant factor in their sculpture, and to enforce this is the chief object of the present communication.

The object of this paper is twofold: in the first place, to classify and sytematise vector differential equations; and in the second place, to show the applicability of quaternions to the theory of differential equations. So far as I am aware, the only paper of importance on the subject is one by Tait. He, however, deals only with simple cases.

In the classification of forms I have followed in general the treatment adopted by Forsyth. Reference will also occasionally be made to Jordan. In what follows the order of the highest occurring differential coefficient will be called the rank of the equation; while the order of the equation will be used, as usual, to denote the order of the equivalent normal system.

In my previous papers on “The Meteorology of Edinburgh” the data referring to local climate were reduced down to the end of 1896. In the time that has elapsed since their publication a number of memoirs from places on the Continent and elsewhere have appeared in which the data, mostly dealing with long periods, were brought down to the end of 1900. In order to facilitate the comparison of the Edinburgh record with those referred to, I have completed new monthly and annual averages for the ten years 1891–1900, the fifty years 1851–1900, and for longer periods, which embrace 137 years in the case of mean temperature, 131 years for mean barometric pressure, 130 years for the non-instrumental phenomena, and 124 years for rainfall (see Tables I. to IV.). The values previously published have been carefully examined, and a number of errors which escaped detection at the time eliminated. A Table of Errata (see Table XXIX.) is appended. Tables V. and VI. show respectively the mean monthly and annual departure from the normal of the mean maximum and minimum temperatures for the fifty years 1851–1900. A few of the values formerly published (see Table XVIIL, Trans., vol. xxxix. pp. 130–133) are about 0°·6 too high, but they have been corrected.

Considering the abundance of this gadoid, it is a matter for surprise that our knowledge of its life-history is not more complete. The work of Prof. Sars upon the cod in Norwegian waters is well known, and need not be referred to in any detail here. It is, however, important to avoid the assumption that his account will apply in every detail to the cod of British waters.

In 1883 Prof. M'Intosh showed that the cod spawned in early spring and that spawning was effected outside the territorial limit. Dr Fulton, from an examination of a number of captured specimens, was led to the same conclusion; and further, that the cod, whilst having an extended spawning period reaching from January to June, must be regarded as having a main period of February, March, and April, the great majority of individuals spawning in the month of March. Thus whilst 40 per cent, of the fish examined in March were mature, the proportion fell to 16 per cent, for February and April, and 10 per cent, for May. In estimating the growth of this species, by the method pursued below, it is probably most correct to regard the true spawning period as including the months of February, March, April, and May. A study of the distribution of pelagic eggs off the Frith of Forth leads to a similar conclusion. From this latter source the spawning period of the cod can be defined as extending from the third week in February to the second week in May, with a maximum in early April.

(1) The persymmetric determinant of the nth order

being such that in every case the element in the place r, s is the same as the element in the place r − 1, s + 1, and therefore having only 2n − 1 independent elements, viz., the elements a1, a2, …., a2n−1 forming the first row and last column, is conveniently denoted by

As it is a special form of axisymmetric determinant, any known relation between minors of the latter must of course hold in regard to the corresponding minors of the former.

Using Bellenden's translation of Boece, he tells us, “Attoure in all the desertis and muris of this realme growis aue herbe, namit hadder, but any seid, richt nutritive baith to beistis and fowlis; specialie to beis. This herbe, in the moneth of July, has ane floure of purpure hew, als sweit as huny. The Pichtis maid this herbe, sum time, ane richt delicus and hailsum drink. Nochtheless, the maner of the making of it is perist, be exterminioun of tlie said Pichtis out of Scotland; fur they schcw nevir the craft of the making of this drink but to thair awin blud.”

Gemünden is situated in Rhenish Prussia, about eighteen miles to the west and slightly also to the south of Bingen, in the district known as the “Hunsrück,” and the rock in which the fishes to be described in this memoir occur is called the “Hunsrück Slates.”

These Hunsrück slates belong to the Lower Devonian of the Rhenish area, and the position assigned to them by German geologists is as follows:—

The above table is quoted from Kayser and Lake's Text-book of Comparative Geology, as is also the following brief statement regarding the Hunsrück slates themselves.

§ 1. In a communication read before the Society, 3rd December 1900, Dr Muir discusses the generalisation, for more than two pairs of variables, of the proposition that: If

then

If we interpret (x, y) and (ξ, η) iis points in a plane, it is manifest that the transformation thereby obtained is a Cremona transformation. It has the special property of being reciprocal or involutive in character; i.e., if the point P is transformed into Q, then the repetition of the same transformation on Q transforms Q into P. Symbolically, if the transformation is denoted by T. T(P) = Q, and T(Q) = T2(P) = P; so that T2 = 1, and T = T−1. Moreover, if the locus of P (x, y) is a straight line, the locus of Q (ξ, η) is in general a conic.

This communication deals with the fossil plants of the Calciferous Sandstone Series and Carboniferous Limestone Series of Eskdale and Liddesdale; of the Lower Coal Pleasures of Canonbie; of the Middle Coal Measures of Byre Burn, Canonbie; and of the Upper Coal Measures, Joekie's Syke, Cumberland; to which is added a list of the fossil plants of the Calciferous Sandstone Series of Cumberland and Northumberland, with the object of comparing them with those from the corresponding series in Scotland.

Since Lord Kelvin showed that the resistance of the magnetic metals was influenced by the state of magnetisation of these substances, a good deal of work has been done in investigating more carefully the extent and nature of the relation. It does not appear, however, that any attempt has been made to discover how change of temperature affects this influence. Twelve years ago in Japan I set one of my Japanese students to the construction of an apparatus suitable for making this research. A few preliminary experiments were made; and leaving Japan shortly after, I brought the apparatus with me in the hope of continuing the work in this country. Not till this summer, however, did I find leisure to return to the problem. The results obtained I wish now to lay before the Society.

The behaviour of metals under stress has long been the subject of investigation, both by mathematicians and physicists, so that the laws of strength are tolerably complete. Owing to the importance of iron and steel in construction, these materials have been subjected to very extensive tests, particularly in simple tension and compression.

Numerous tests of cylindrical iron and steel bars in torsion are also available, the bulk of these being tests to destruction of samples of material used in actual machines and structures designed by engineers. In such tests scientific accuracy is not of much importance, the chief consideration being the obtaining of sufficient data for use in design. The most accurate torsional work upon iron and steel has been the work of physicists, and nearly all their investigations have been conducted upon specimens of very small sectional area; the reasons for this, no doubt, being that such specimens in the form of wires are easily obtainable, and of great uniformity in size and quality, while large test pieces are costly to prepare, and, moreover, cause considerable difficulty in testing, because of the magnitude of the forces involved. Owing to the mode of manufacture, the physical properties of wire often differ to a considerable extent from turned specimens of iron and steel. These differences may be caused by the hardening effect of the drawing, minute cracks in the wires, want of roundness, and the like. It therefore appeared probable that experiments on the lines indicated by physicists would be of some service, and it was with this idea that the investigation was commenced.

(26) Of the various determinant forms thus far obtained the most promising is that of §8 or that of §14; and to these it is desirable now to return in order to obtain an expression for the eliminant in the ordinary non-determinant notation. In doing so it will also be well to make a slight change in the coefficients of the three quadrics, viz., to write f, g, h for 2f, 2g, 2h, as in this way the diversity in the cofactors of the determinants occurring in the last three rows of either form of the eliminant disappears.

Up to the present time no systematic account of the cranial characters of the people of Scotland has been published. Incidental references to, and measurements of, a limited number of Scottish skulls may indeed be found in the writings of various authors, as in Sir Daniel Wilson's Prehistoric Annals of Scotland, in Drs Davis and Thurnam's Crania Britannica, and in Professor Cleland's Memoir on Variations in the Human Skull. Measurements of five Scottish crania are recorded by Sir W. H. Flower in the Osteological Catalogue (Man) of the Museum of the Royal College of Surgeons of England, four of which were found amongst the ruins of an ancient Culdee Monastery at St Andrews, and the fifth is said to be that of a Highlander. Dr Barnard Davis, in his Thesaurus Craniorum, gives the measurements of a somewhat larger number, six of which were from Caithness, and one is stated to be a Scottish Highlander. The same skulls have been remeasured and described by Dr J. G. Garson.

1. The present paper deals with a, series of experiments made in the Physical Laboratory, Edinburgh University, from January to October 1899, with the object of determining the convective loss of heat from a cooling body owing to the action of currents of air.

In the early part of 1900 I received from J. H. Ponsonby, Esq., a small collection of land molluses from N.W. Borneo, with a request that I would investigate and report upon the same. This collection, the property of the Sarawak Museum, proved, on examination, to contain examples of two new genera (Wiegmannia and Isselentia) of great interest. In the case of the former genus there were only two specimens, each belonging to a different species; and as more material was very desirable, Mr Ponsonby very kindly invited the authorities of the Museum to send over, if possible, a further collection. This, in due course, arrived, but contained duplicates of Isselentia only, the remaining specimens being all new species, excepting one, which proved to be the Damayantia dilecta of Issel, which, I believe, has not previously been found since described in 1874.

The Canonbie Coalfield occupies a small tract of ground between the Liddel Water and the river Esk in the south-east part of the county of Dumfries. Though of limited extent, the coalfield has aroused considerable interest, due partly to the important series of plants obtained from the beds, and partly to the questions bearing on the correlation of the Carboniferous rocks of the Scottish Border with those in the North of England and Central Scotland.

Introductory.—Among the toothed whales (Odontoceti) the porpoise is the best-known representative of those members of the genus Delphinus or true dolphins which present a rounded muzzle as distinguished from a snout, and consequently it has already frequently been subjected to anatomical examination of a more or less detailed character. As in the case of all the Cetacea, however, the rapidity with which decomposition affects the various tissues and organs has hitherto proved a barrier to a prolonged and systematic examination of them, while, even under the most favourable conditions, the increasing putridity of the carcase has seriously militated against the recording of accurate observations.

(1) This is a subject to which very little study has been directed. The first to enunciate any proposition regarding it was Jacobi; but the solitary result which he reached received no attention from mathematicians,—certainly no fruitful attention,—during seventy years following the publication of it.

Jacobi was concerned with a problem regarding the partition of a fraction with composite denominator (u1 − t1) (u2 − t2) … into other fractions whose denominators are factors of the original, where u1, u2, … are linear homogeneous functions of one and the same set of variables. The specific character of the partition was only definable by viewing the given fraction (u1−t1)−1 (u2−t2)−1…as expanded in series form, it being required that each partial fraction should be the aggregate of a certain set of terms in this series. Of course the question of the order of the terms in each factor of the original denominator had to be attended to at the outset, since the expansion for (a1x+b1y+c1z−t)−1 is not the same as for (b1y+c1z+a1x−t)−1. Now one general proposition to which Jacobi was led in the course of this investigation was that the coefficient ofx1−1x2−1x3−1…in the expansion ofy1−1u2−1u3−1…, where

is |a1b2c3…|−1, provided that in energy case the first term of uris that containing xr.

(1) The simplest possible case of the development referred to in the title is one which is seen to follow instantly from the determinant definition of a Pfaffian. Thus, by the said definition,

and therefore

the result reached being that a Pfaffian with certain zero elements is expressible as a determinant. It should be noted that, as in many instances where this form of definition is used, there is an ambiguity as to sign : it is of less importance to note that here the appropriate sign is –.