1. The integral occurs so frequently in various branches of research that, as far back as 1783, Laplace suggested that it would be useful to tabulate its values for successive ranges of integration. It is employed in investigations on the theories of refraction, conduction of heat, of errors of observation, of probabilities, etc. These are familiar to physicists and need not be dwelt upon.

In the year 1819 the University of Edinburgh acquired by purchase a large zoological collection from M. Dufresne, of Paris, including a cabinet containing upwards of 12,000 specimens of insects. Some years later (in 1855), when the whole of the University collections were formally transferred to the Science and Art Department, the Dufresne cabinet became public property.

While consulting lately the volume by Godart devoted to the article “Papilio” in the famous Encyclopédie Méthodique, I was surprised to see a reference to the “Dufresne cabinet” in one of the descriptions of new species, and it immediately occurred to me that the specimen in our possession must be the actual “type” of the species, which discovery led me to search carefully through the whole of the volumes devoted to insects in this well-known work, with a view to finding as many types as possible. At the same time, I thought it possible that Olivier, who wrote his great Histoire Naturelle des Insectes—Coléoptères about the same time, might also mention some of Dufresne's specimens, and this I found to be as I had anticipated.

The former paper (5) on the Anatomy of Cephalodiscus was incomplete in certain particulars, which I am now in a position to add. These are mainly in regard to the alimentary canal, the nervous system, and the reproductive organs. To the present description of these will be appended an account of the budding processes, which are a very conspicuous feature of the species.

The existence of a series of trap dykes cutting the Old Red Sandstone strata of the Orkneys has been noted by most of the geologists who have examined the islands. In his description of the Mainland or Pomona, Professor Jameson states (I., p. 233, vol i.) that “at Yesknaby is the only basaltic rock which I observed in the whole island. It forms veins which traverse the common argillaceous sandstone. The crystals of hornblende, which are contained in it, are larger than usual in such rocks, being more than an inch long and half an inch broad. I sometimes observed small cavities filled with bitumen.” Sir Archibald Geikie, in his account of the Old Red Sandstone of Orkney, remarks (II., p. 408): “Here and there a few basalt dykes—far outlying portions, no doubt, of the great Tertiary series of the West of Scotland—cut through the flagstones with a prevalent direction towards west or north-west.” Messrs Peach and Horne, in their paper on The Old Red Sandstone of Orkney (III., p. 14), describe them in the following terms: “Several dykes of basalt were observed among the islands. They are most numerous and conspicuous on the west coast of the Mainland from Breckness to Skaill, but as they have been so often described, it is unnecessary to refer to them in detail. They have the same lithological characters, and behave in exactly the same manner as the dykes in other parts of Scotland, which have been regarded as the product of volcanic energy in Miocene times. A noticeable feature about the Orcadian representatives is, that they are usually divided up the centre of the dyke by a line of vesicles. This is not an uncommon feature elsewhere.” In the chapter on the Geology of Orkney in Tudor's The Orkneys and Shetland, by the same authors, the dykes are referred to in similar terms (IV., p. 191). A somewhat more minute examination of these dykes was made by Professor M. Foster Heddle. He notes the presence of augite, olivine, and hornblende in certain dykes near Skaill (V., p. 118), and gives a map of the dyke which cuts the west end of the granite outcrop at Inganess (V., pl. viii.). He figures also a crystal of augite of simple form which he found in a dyke on Scabra Head, Rousay (V., p. 128).

In a collection of Indian butterflies purchased by the Edinburgh Museum in the year 1890, I found a specimen of Hestina which is of some interest. After carefully examining it, and comparing it with a figure by Oberthür of a form which he regards as a melanic aberration of the well-known Hestina nama, Doubleday, I have come to the conclusion that our example is another and more melanic form of the same species. From Oberthür's figure, which he calls ab. melanina (Études d'Entomologie, xx., 1896, p. 30, pl. 10, no. 177), the present specimen differs in the fore wing, in the absence of the inner row of whitish spots, in the size and position of the upper spots in the outer series, and in the spot at the anal angle being nearer to the base of the wing. At the base of the cell there is a rich cream-coloured sub-triangular spot, instead of the slender greenish streak. There is no ferruginous border to the hind wing, and the whitish streaks are more elongated, and in the anal half of the wing only very faintly indicated.

1. The existence of relations between the coaxial minors of a determinant was first discovered by MacMahon in 1893. The whole literature of the subject is comprised in three papers, viz.:—

MacMahon, Phil. Trans., clxxxv. pp. 111–160.

Muir, Phil. Mag., 5th series, xli. pp. 537–541.

Nanson, Phil. Mag., 5th series, xliv. pp. 362–367.

My present object is to continue the investigation of the relations in question, and more particularly to draw attention to an explicit expression for a determinant of the 4th order in terms of its own coaxial minors. At the outset some fresh considerations regarding determinants in general will be found useful.

The immediate purpose of the present research was the investigation of the temperature at different pressures of boiling mixtures of steam and salts, analogous to the well-known freezing mixtures of ice and salt.

When steam is blown through common salt in coarse powder, it condenses to water, which dissolves some of the salt, and the resulting brine is kept boiling by the arrival of more steam. The temperature of this boiling mixture is quite constant so long as there is an abundant supply both of steam and of salt, and as the atmospheric pressure does not change, it is about 8·5° C. above the temperature of boiling water when the barometric pressure is the normal of 760 mm. When the barometric pressure is 560 mm. this excess has fallen to 8·0° C. Most other salts behave in a similar way.

The unusually fine stem which forms the subject of this paper was found by Mr T. Kerr, of Edinburgh, in rocks of Calciferous sandstone age exposed in a railway cutting at Dalmeny, Linlithgowshire. Mr James Bennie, of the Geological Survey of Scotland, called the attention of Mr Kidston to the fossil, and the latter very generously placed part of the specimen in the hands of one of us for investigation.

I have to apologise for placing this communication before you in a somewhat incomplete form. My work has been much interrupted in the past, and it will again shortly be stopped for a considerable time. I have therefore decided to place before you the results so far as they are worked out.

The earlier mineralogists laboured under two great disadvantages. They could not readily, on account of the small number of students of chemistry, call in the aid of that science: and at the time when mineralogy was becoming a distinct branch of science chemistry was in itself crude as well as cumbrous. They were thus forced to rely chiefly upon external properties; and, where crystalline form was absent, they were confined to what may be called physical properties alone.

Their knowledge of the composition of bodies being thus limited and uncertain, the old nomenclature was to a considerable extent founded upon external features alone.

In 1892 I communicated to this Society an account of a collection of Eurypterids from the Upper Silurian rocks of the Pentland Hills, in which I described new species and a new genus. Since that time I have had the opportunity of examining two other collections from the same locality. One of these was formed with the aid of a grant from the British Association. The fossiliferous bed—for these remains occur in a single thin bed of rock—was laid bare, and a considerable amount of it removed and split up. In this latter work I was fortunate to get the assistance of Mr John Henderson, the original discoverer of the locality, and I am glad to have this opportunity of recording my indebtedness to him. A fair number of specimens was thus procured, which not only threw considerable light on the structure of some of the already described forms, but gave evidence of some species as yet undescribed. More important than this collection, however, was the one formed by the late Mr Hardie of Bavelaw Castle, who had for many years collected among the Pentland Silurians. His collection was, on his death, acquired by the Edinburgh Museum of Science and Art, and Dr Traquair kindly put the collection in my hands for examination. From the size of the collection I expected some new and interesting results, and have been far from being disappointed. Seven new species of Eurypterids have come to light and one Scorpion.

In the autumn of last year Sir Archibald Geikie, F.R.S., Director-General of the Geological Survey of the United Kingdom, Kindly placed in my hands for determination a number of specimens of Cephalaspis, collected by his officers in the Lower Old Red Sandstone of the neighbourhood of Oban. On examining them, I found that they all belonged to one species, which was, however, new to science.

Accordingly I drew up a brief diagnosis of this new form, which was included by Sir Archibald in his Summary of Progress of the Geological Survey for 1897, and it is now my privilege, with his sanction, to offer to this Society a more detailed description of the species, accompanied with figures.

The purpose of this paper is to discuss the conditions under which the C discriminant of a system of curves furnishes a curve which at every point of its length is touched by a curve of the system.

The experiments described in this paper were commenced with the view, not only of determining the absolute thermal conductivity of nickel, but also of comparing the results found by Forbes's and Ängstrom's methods for the same specimen. Although some readings were taken for Ängstrom's method, that part of the investigation was not completed, because it was found that the experimental errors — unavoidable, on account of the necessity of measuring rapidly changing temperatures—would be too great for the results to be of any value. The thermal conductivity of a portion of the bar of nickel used for Forbes's method was determined by a direct method involving the determination only of steady temperatures, and the results so obtained are given in the latter portion of this paper.

Among the specimens from the Yorkshire Coal Field which have been collected by Mr W. Hemingway, Barnsley, and submitted to me for examination at various times, are the remains of several cones which are referable to the genus Sigillariostrobus, Schimper.

Notwithstanding the great frequency of the genus Sigillaria in the Coal Measures, and especially in the Middle Coal Measures, examples of their fructification are very rare. This is the more remarkable, as specimens of Sigillaria, showing cone scars, though not common, are occasionally met with. Possibly, however, the apparent rarity of Sigillarian cones is due, in part, to our inability at present to distinguish them in all cases from cones generally placed under the name of Lepidostrobus, which latter genus there is every reason to believe comprises cones that belong to several genera of Lycopods.

As the result of two visits to Orkney, in which he was accompanied by Mr B. N. Peach, he pointed out that the yellow sandstones of Hoy did not pass down conformably into the flagstones which form the basis of that island, but were separated from them by a marked unconformity. At the base, of the upper sandstones lay a series of contemporaneous lavas and ash beds, which were in all probability erupted, from certain ‘necks’ in the low-lying district at the foot of the Hoy Hills. These rocks he regarded as belonging to the upper Old Red Sandstone. The lower Old Red Sandstone consisted principally of a great thickness of flagstones, with which were interstratified beds of yellow and red sandstone, and occasionally of conglomerate. The fossils belonged exclusively to this lower series; and a table is given, compiled by Mr C. W. Peach, showing the distribution 9f fossil fishes in the lower Old Red Sandstone of Lake Orcadie, including those of Orkney so far as known at that time. As Sir Archibald Geikie anticipated, subsequent revision has necessitated “considerable pruning of the fossil lists.” The conglomerates around the granite axis of Stromness formed merely a local base, “due to the uprise of an old ridge of rock from the surface of the sheet of water in which these strata were accumulated,” and were presumably not on the same horizon as the thick conglomerates on which, in Caithness, the lowest flagstones rest. The sandstones interbedded with the flagstones in South Ronaldshay were regarded as in all probability the northward continuation of the similar rocks at Gill's Bay, Huna, and John o'Groats, on the south side of the Pentland Firth. From a geological point of view, the brief notice of the Old Red Sandstone of the Orkneys contained in this paper forms by far the most important contribution to the knowledge of the subject published up to that time.

The remarkable fossil fish which forms the subject of the present communication belonged originally to the late Mr Powrie of Reswallie, and was acquired, along with the rest of his collection, by the Edinburgh Museum in 1892.

The specimen was obtained by Mr Powrie in a quarry of the Lower Old Red Sandstone at Turin Hill, near Forfar.

In his monograph on the Fossil Cephalaspidœ of Great Britain, published in 1870, (p. 41), Professor E. Ray Lankester refers in the following terms to the specimen in question:—

“A very remarkable specimen of shagreen-like structure has been discovered by Mr Powrie in Forfarshire, in beds which have furnished Cephalaspis. As I have not been able to assign it to Cephalaspidian fishes, though it may possibly, be connected with them, I only allude to it here. It consists of a surface covered with minute spinous tubercles, the whole having the appearance of a fossilised piece of shagreen, and its shape is more or less that of a Cephalaspid, with head and body.