§1. As a preliminary to the establishment of the law in question, it is necessary to state and exemplify another law to which I have elsewhere directed attention, viz.,

THE LAW OF COMPLEMENTARIES.

To every general theorem which takes the form of an identical relation between a number of the minors of a determinant or between the determinant itself and a number of its minors, there corresponds another theorem derivable from the former by merely substituting for every minor its cofactor in the determinant, and then multiplying any term by such a power of the determinant as will make the terms of the same degree.

§1. It is well known that by a simple transformation of a determinant we may cause a zero to take the place of any one of the elements. The theorems of Hermite and Horner, for example, for depressing the order of a determinant may each of them be viewed as the result of repeated transformations of this kind, the operation being continued until all the elements of a row or column except one are replaced by zeros.

I am indebted to the kindness of Professor Ramsay, Director-General, and of Professor Geikie, Director of the Scottish Branch of the Geological Survey of Great Britain, for the privilege of examining and describing a remarkable collection of fossil fish-remains from the Lower Carboniferous rocks (Calciferous Sandstone Series) of Eskdale and Liddesdale. Most of the specimens were collected by Mr Arthur Macconochie, one of the collectors attached to the Scottish Geological Survey; and Mr Walter Park of Brooklyn Cottage, Langholm, has also willingly co-operated in the search, so far as the district of Eskdale is concerned. I have myself also had the pleasure of twice visiting Eskdale, along with Mr Macconochie and Mr B. N. Peach, and on these occasions I obtained a few specimens for my own collection.

By the permission of Professor A. C. Ramsay, LL.D., F.R.S., Director-General of the Geological Survey of Great Britain and Ireland, and Professor Geikie, LL.D., F.R.S., Director of the Geological Survey of Scotland, I have been permitted to describe several new Crustaceans which have come under my notice in my capacity of Acting Palæontologist to the Scotch Survey. They are from the cement-stone group of the Calciferous Sandstone series of the Scottish border, and, with a very few exceptions, were got from one locality on the river Esk, about four miles south of Langholm, in Dumfriesshire, and were almost all collected by A. Macconochie, Fossil Collector to the Geological Survey of Scotland. They belong to two orders, viz., Phyllopoda and Decapoda.

Of all the various spectra which the progress of experimental science has enabled man to observe in the present day, none are so rich, varied, and important, as those of gases. And no wonder! for it is only when matter has been reduced to the gaseous condition, that it is able to specialise itself and write its character with much of its history in any otherwise smooth, undefined, continuous spectrum; while, if in former times, men would have found it an impossibility to drive many of the more refractory substances into the state of incandescent vapour, what is there now anywhere on the surface of this earth which, in small quantity, can resist the action of a powerful and condensed induction spark of electricity; and what application of that spark is so neat, elegant, convenient, and economical, as when it is employed in conjunction with so-called gas-vacuum tubes.

If Sn be a rational integral function of x of the nth degree, and Sn-1 Sn-2… S1 S0 a series of such functions of the n — 1th, n—2th, &c., degrees, so related to Sn that, when any one of the whole series S0 S1…. Sn vanishes, the two on opposite sides have opposite signs, and farther Sn-1 and Sn have always opposite signs when x is just less than any real root of Sn = 0, then S0 S1… Sn-1 may be called a set of Sturmians to Sn. It is obvious that the problem of finding such a set of functions admit of an infinite number of solutions. The first discovery of such a set was made by Sturm, and the researches of Sylvester, Hermite, and others have shown how other solutions of the problem may be obtained.

In a paper by Mr E. W. Binney on the Fossil Fishes of the Pendleton Coal Field, published in 1841, the dentary bone of Rhizodopsis is figured as the “upper jaw of a new species of Holoptychius,” to which, however, he did not attach any specific name. In the same paper its scales are also figured and referred to the same genus. Scales belonging to the same fish were afterwards figured by Professor Williamson under the name of Holoptychius sauroides, and again by Mr Salter, as those of Rhizodus granulatus. Both of these specific names occur under Holoptychius in Agassiz's general list of Ganoids published in 1843, but as they were unaccompanied either by figures or descriptions, it is really immaterial which of them, if indeed either, was applied by him to the fish in question. The authority for the term “sauroides” as applied to the common species of Rhizodopsis, the only species of the genus which is as yet known with certainty, must therefore remain with Professor Williamson. Holoptychius sauroides of Binney and of Messrs Kirkby and Atthey is quite another fish, now also distinguished generically as Strepsodus, and for it the specific name “sauroides” is therefore equally valid.

The phosphines, or substances derived from phosphuretted hydrogen by the partial or complete replacement of its hydrogen by hydrocarbon radicals, have formed the subject of many valuable researches; but although their discovery was anterior to that of the compound ammonias, their study has made comparatively little progress. This is no doubt mainly due to the difficulty attending their preparation, a fact which is immediately forced upon the notice of any one who wishes to investigate them.

In this paper I give an account of observations made during a visit in 1879 to the Færöe Islands in company with my friend Mr Amund Helland of Christiania. The principal object of our journey was to examine the glacial phenomena of the islands, but we studied so far as we could the various rockmasses of which the group is composed, and constructed a geological sketchmap to show the line of outcrop of coal, the disposition of the strata, the direction of dykes, and the trend of the glaciation. I have only to add, that all the observations recorded in the following pages were made in concert with my friend, and I am glad to say that we were quite at one in our general conclusions.

At the surface of separation of any two different substances in contact, there exists in general an electromotive force tending to maintain a certain difference of potential between them. This principle, established for metals by Volta in 1796, has been extended by later investigators to other substances, including liquids and gases. From these early researches of Volta, and the later more elaborate inquiries of Kohlrausch, Hankel, and Gerland, there have been deduced certain fundamental laws, which have been fully corroborated by the recent work of Clifton, and Ayrton and Perry. If, of a number of conductors set serially in contact, the difference of potential between each successive pair is quantitatively estimated and reckoned positive or negative, according as the first member of the pair is at a higher or lower potential than its successor, then the difference of potential between the first and last members of the chain is equal to the algebraic sum of the potential differences between the successive contiguous pairs.

In a paper by Professor Crum Brown and the author on Dimethyl- Thetine and its Derivatives, attention was drawn to the analogies which frequently exist between elements which have different atomicities, and which are usually considered as belonging to different families. The most striking examples of such elements are boron and carbon, gold and platinum, and phosphorus and sulphur.

Water is perhaps the most abundant and most universally distributed form of matter on the earth. It has to perform more varied functions and more important duties than any other kind of matter with which we are acquainted. From its close connection with all forms of life, it has been the subject of deepest interest in all ages. It is constantly changing from one of its states to another. At one time it is solid, now liquid, and then gaseous. These changes take place in regular succession, with every return of day and night, and every successive season; and these changes are constantly repeating themselves with every returning cycle. Of these changes, the one which perhaps has the greatest interest for us, and which has for long ages been the subject of special observation, is the change of water from its vaporous state, to its condensation into clouds, and descent as rain. Ever since man first “observed the winds “and “regarded the clouds,” and discovered that “fair weather cometh out of the north,” this has been the subject of intensest human interest, and at present forms one of the most important parts of the science of meteorology, a science in which perhaps more observations have been made and recorded than in all the other sciences together.

The object of this investigation was to obtain information as to the change of specific resistance produced in wires of various metals by different amounts of elongation. The present paper refers to experiments on copper, iron, and German silver wires.

The structures known as pedicellariæ were first described by O. F. Müller. He regarded them as parasitic organisms, and described under the generic name of Pedicellaria three species—Pedicellaria tridens, P. triphylla, and P. globifera. Lamarck, in the first edition of his “Animaux sans Vertèbres,” published in 1801, and also in the second edition, following Müller, classified the genus Pedicellaria with the Corynidæ and Hydridæ among the naked polyps. Cuvier also, in the first edition of the “Regne Animal,” took the same view of the nature of pedicellariæ. It was not until 1825 that the real nature of these organs was recognised. Delle Chiaje, in his work on invertebrate animals, asserted that pedicellariæ formed an integral portion of the Echinus, and were simply appendages of the test. The following quotation from his work shows that he thoroughly understood the true nature of these structures. He says (vol. ii. p. 324):—“Fanno elleno parte integrale degli echini e servon loro per attacarsi a corpi adiacenti ed anche a ritenere gli animaletti da cibarsi.”

In the progress of the Geological Survey of the South of Scotland, specimens referable to the genus Eoscorpius have been gradually accumulating. In 1876 J. Bennie, Fossil Collector to the Survey, obtained an example from the Coal-measures of Fife. Since then fragments have been disinterred by him and by A. Macconochie, also Fossil Collector to the Survey, from the Calciferous Sandstone series in the counties of Edinburgh, Berwick, Roxburgh, Dumfries, and Northumberland and Cumberland. It was not till the spring of last year (1880) that they began to be found in such a state as to necessitate a description of the fossils. In the summer of that year A. Macconochie obtained an almost entire example from the neighbourhood of Langholm, in Dumfriesshire. This year (1881) J. Bennie has secured several good though fragmentary specimens from the neighbourhood of Edinburgh, while A. Macconochie has sent in several from the counties of Berwick and Northumberland. In my capacity of Acting Palæontologist, I have had an opportunity of studying these remains, and by the permission of A. C. Ramsay, LL.D., F.R.S., Director General of the Geological Survey of Great Britain, and Professor A. Geikie, LL.D., F.R.S., Director of the Geological Survey of Scotland, I have been allowed to describe them.

In the following communication an account is given of some experiments made in the Physical Laboratory of the Glasgow University, whose object was the measurement of the change of conductivity produced by strain in a metal conductor.

Two constants are required in order to determine this alteration in the most general case of a homogeneous isotropic conductor, strained in any way, as may be easily inferred from the analogy with elastic strains and stresses, constants expressing the change produced by uniform cubical compression and by simple distortion. The determination of the latter for brass is the object of the present paper.

I had added (at p. 140 of my paper on “Gaseous Spectra”) to the “small dispersion” account therein given of the above mentioned spectrum of luminous oxygen gas,—that two, if not four, of its very few and scanty lines appeared, when viewed with much higher dispersion, to be double; but that I hoped to give a more exact account of them, after completing some arrangements then in progress for increasing both the dispersion and magnifying power of my spectroscope.