It is a fundamental principle in Algebra, that a problem admits of solution, when the number of independent equations is equal to that of the unknown quantities. If simple expressions only occur, the answers will always be found in numbers, either whole or fractional. But if the higher functions be concerned, the values of the unknown quantities will commonly be involved in surds, which it is impossible to exhibit on any arithmetical scale, and to which we can only make a repeated approximation. Hence the origin of that branch of analysis which is employed in the investigation of those problems, where the number of unknown quantities exceeds that of the proposed equations, but where the values are required in whole or fractional numbers. The subject is not merely an object of curiosity; it can be applied with advantage to the higher calculus. Yet the doctrine of indeterminate equations has been seldom treated in a form equally systematic with the other parts of algebra. The solutions commonly given are devoid of uniformity, and often require a variety of assumptions. The object of this paper is to resolve the complicated expressions which we obtain in the solution of indeterminate problems, into simple equations, and to do so, without framing a number of assumptions, by help of a single principle, which, though extremely simple, admits of a very extensive application.

It is my intention in this, the second part of my memoir on the Craniology of the Races of India, to give the results of my examination of skulls obtained from the districts occupied by the aboriginal tribes in Chúta Nágpúr, the Central Provinces, the people in the province of Orissa, and to compare them with the skulls of some other aboriginal people.

The majority of the specimens described belong to the Indian Museum, Calcutta, and through the courtesy of the Trustees I was permitted to have them on loan for purposes of study. Many of these crania had been those of persons who had died in jail. The names, tribes, and castes, and not unfrequently the age, stature, and other physical characters, had been recorded in the prison books, and were embodied in the lists which were sent to me along with the skulls by the authorities of the museum. Several of these skulls were especially interesting, as having been presented to the museum by Colonel Dalton, the author of the valuable treatise on the Ethnology of Bengal, Other specimens in the museum had been obtained from the Medical College, Calcutta, and several were presented by Professor D. B. Smith; in all probability they were from bodies which had been used for anatomical purposes. Mr W. H. P. Driver also had presented a series of crania from Ranchi.

Of all known examples in physical science, of simplifying, and at the same time “precisionising” some of its fundamental data, which might otherwise fall to be entangled in high numbers, none has been happier than Fraunhofer's application of the letters of the alphabet to certain chief lines in the solar spectrum. Happy both in its conception by the inventor, and its universal acceptance since then by the world. Whence it comes to pass now, that in every country, whoever observes the solar spectrum at all, with whatever instrument, large or small, diffracting or refracting, and whether he holds to the undulatory, or any other theory of light, and catalogues spectral lines either in Wave-lengths or Wave-numbers, or merely in terms of the brass scale screwed to his instrument by a maker,—yet whenever he speaks of the line A, or B, or C, or any other so named by Fraunhofer, he singles out thereby from among thousands, exactly the same identical line which any and every other spectroscopist alludes to under the same simple letter.

It is well known that although the overwhelming majority of specimens of fircones exhibit one or other of the simple spiral arrangements represented by the terms of the ordinary series ½, ⅓, ⅖, ⅜, , &c., whose generating and successive secondary spirals are indicated by the numbers 1, 2, 3, 5, 8, 13, &c., yet exceptional cases occur now and again, where we find either conjugate spirals of the ordinary system, or arrangements (usually simple, but sometimes conjugate) belonging to other systems of spirals. Of these exceptional arrangements, perhaps the most common are bijugates of the ordinary system, giving the numbers 2, 4, 6, 10, 16, 26, &c., and simple spirals belonging to the system ⅓, , &c., giving the numbers 1, 3, 4,7,11, 18, &c. More rarely, trijugates of the ordinary system occur, giving the numbers 3, 6, 9, 15, 24, 39, &c.; or spirals of the system , ⅕, , &c., giving the numbers 1, 4, 5, 9, 14, 23, 37, &c.; not to speak of various other arrangements, some of which will fall to be considered in the special cases which form the subject of the present communication.

The following experiments, in continuation of our research on the discharge of electricity, were made in Professor Tait's Laboratory during the College Session 1878–79, and papers briefly describing the results were read before the Society on 6th January 1879 and 19th July 1880.

In summer 1776, Professor Ferguson observed a particular appearance on the hill of Arthur's Seat, near the summit, which drew his attention, and which he could not understand. He then carried Dr Black and me to the place, where we found something which, at a distance, resembled the withered grass of a foot-path, but which traversed a shoulder of the hill, in such a direction as corresponded to neither sheeptrack nor foot-path. Upon a near inspection, it appeared to be a narrow stripe of the grass quite dead and withered. The breadth of this stripe was about nine, or, in some places, twelve inches; the sides of this track were perfectly defined, without any gradation from green to withered grass, all the plants in the track being killed, without the contiguous part having suffered in the least. The length of this track was considerable, a hundred yards or two, extending from the south-east side of the southmost hill through a hollow, and ascending obliquely the shoulder of the summit of Arthur's Seat on the south-east side.

Authorities on the action of medicinal substances agree in including convulsions among the effects on man of belladonna, and of its active principle, atropia. Similar effects are described as occurring when large doses of these substances are administered to dogs, rabbits, and other mammals, and to various birds. The recent remarkable progress of our knowledge of the exact and ultimate physiological action of many medicinal substances is greatly due to investigations that have been made on animals of a lower type of organisation; and, accordingly, numerous observers have instituted experiments with atropia on such animals, and especially on frogs. Hitherto, however, convulsions and tetanus have not been described among the effects of atropia-poisoning in cold-blooded animals.

In March 1869 I communicated to the Society a paper entitled “Motion of a Palladium Plate during the Formation of Graham's Hydrogenium,” which appears in the Proceedings for Session 1868-69. When engaged with this subject many points of interest regarding the behaviour of palladium containing occluded hydrogen, suggested themselves for investigation, and in concluding the paper I remarked “ that careful determinations must be made of the electromotive force, latent heat, &c., of hydrogenium ” before we could arrive at any conclusion regarding the condition of the absorbed hydrogen. Subsequently Professor Tait made a series of determinations on the “ Electrolytic Polarisation of Palladium Electrodes,” devising a new and ingenious method for the purpose. Although at different times subsequent to my first communication, the problem of determining the physical constants of hydrogenium recurred, as my attention was in the meantime directed to the specific heat of carbon at high temperatures, no progress was made with the investigation until September 1872, when the results of my preliminary experiments were communicated to the “ Philosophical Magazine,” under the title, “ Note on the Specific Heat of Hydrogenium.” In that note it is stated that by means of a specially constructed calorimeter the specific heat of hydrogen in palladium is found to be 3.1 per atomic weight, nearly identical with that of gaseous hydrogen. The present paper deals with some of the physical constants of hydrogenium, more especially with the specific gravity, specific heat, and co-efficient of expansion.

The very curious Parasitic Crustacean, the structure and larval form of which we propose to consider in the following communication, we obtained in the month of August last, in considerable numbers, in the branchial pouches of a Lophius piscatorius, which we captured at Whitby in Yorkshire.

At the suggestion of Prof. Tait, an attempt has been made in this paper to apply the method used by him in § 21 of his paper on “The Foundations of the Kinetic Theory of Gases” to a case of the question of the distribution of energy in a system of non-homogeneous impinging spheres.

The problem may be stated as follows:—Given a very great number of smooth elastic spheres, equal and like in all respects, whose centres of figure and centres of inertia do not coincide, and the sum of whose volumes is but a small fraction of the space in which they move, it is required to find the ultimate distribution of energy among the various degrees of freedom when by collisions the system has attained a “special state.”

The general proposition in the theory of commensurables is to determine the conditions under which lines, surfaces, or solidities, connected with prescribed figures or forms, may have their ratios expressible by integer numbers.

The attention of geometers must have been drawn to this subject by the contemplation of incommensurable lines: the altitude of an equilateral trigon is incommensurable with the base; the diagonal of a square incommensurable with the side, and so on. And, when the two sides of a right angle are expressed by two numbers, the hypotenuse is, in the great multitude of cases, incommensurable with the sides: thus, if the sides be 5 and 7 inches respectively, the length of the subtense cannot be accurately expressed either in integers or fractions. However, when the sides are 3 and 4 units, the hypotenuse is exactly 5 of the same units.

Multiplices Solariorum species, genus testaceorum constituentes, ad Turbinacea Lamarck, ad Gasteropoda Cuvier, pertinens, quæ in geologicis Italiæ stratis reperiuntur, monographico modo investigare fortasse præstat. Ratio nec ambigua; variæ enim dubitandi rationes vigebant, tum circa numerum et nomina specierum, tum circa earundem citationes, et auctorum dissensiones, nec non potissimum circa relationes specierum quæ in Italicis divisionibus supracretaceæ formationis adsunt, sive in se spectatis, sive prout cum viventibus conveniunt aut discrepant.

It is not a little remarkable, that in an age which assumes to itself the character of devotion to science, in anatomy, a science which embraces the best interests of humanity, this question should remain unanswered; What is the meaning of the nerves of the spinal marrow being in regular order and perfectly symmetrical: whilst the ten nerves arising from the brain present no similarity one to another, and agree neither in origin, size, nor distribution?

It is plain that we must be in the dark, not only with respect to the knowledge of the nervous system, but of the animal frame generally, whilst such a question is open and courts inquiry, and yet remains without an effort being made towards its solution. We must, I fear, attribute this neglect in part only to the difficulty of the inquiry, and much to the indifference to all that does not tend directly to profit; on which account it has the better demand on the attention of a learned and philosophical Society.

The investigation of which I am now about to give an account, was undertaken with the view of determining whether the structure of the Torbanehill mineral was similar to or unlike that of coal. I was aware that the subject would be brought before a court of law, and that many scientific persons of great eminence had already spent much time in the inquiry. With the understanding, therefore, that my evidence, should it be required, was to be limited to the structure of coal and of the mineral in question, I gave directions to Mr Bryson, the optician, of this city, to make thin sections of attested specimens of various coals and of the mineral, conceiving that a careful examination of them would easily determine the point. It was soon apparent, however, that a far more extended series of researches was necessary than I at first anticipated; but as it was also evident, from the marked structural differences which were observed in the sections, that the investigation would not be destitute of positive results, I determined on pursuing it to a conclusion.

Permian fishes and their isolated microremains are known from many localities in the Zechstein Basin. However, up to date the vertebrates have never been revealed in the southeasternmost part of this ancient sea. The new material consists of euselachian-type dermal denticles,?Listracanthus sp. dermal denticle,?Omanoselache sp. tooth, actinopterygian scales and actinopterygian teeth. Here, the detailed study of euselachian and actinopterygian remains, their stratigraphic distribution and geographical contexts is presented. Based on the qualitative analysis of teeth shapes several ecomorphotypes were described as well as the probable dietary preferences of fishes were reconstructed. These finds confirmed existence of small predators who fed on soft bodied prey as well as durophagous forms which were feeding on small shelly crustaceans or molluscs. The analysis of stratigraphic distribution of microremains, and their comparison with neighbouring sections revealed a spatially correlatable trend in increasing abundance of fishes in the more clayey parts of sections, interpreted to be positively associated with a sea level transgression event.

In a former paper I described the variation of the fecundity of women according to age, and arrived at the conclusion that the climax of fecundity in women was at or near the age of 25 years. Researches, completed since that paper was read, regarding the variations of length and weight of children according to the mother's age, and regarding the mortality of childbed as influenced by the mother's age, have been published in the “Edinburgh Medical Journal.” The results of these investigations seem to illustrate and confirm the statement made as to the age of the climax of fecundity, for I have found that, about that age, women produce the bulkiest children, as measured by length and weight; and about the same age of the mother there is the smallest mortality in childbed. As still further adorning the age of 25, I may add, that several sets of observations, including some made in St George's-in-the-East, London, and published in the eleventh volume of the “Journal of the Statistical Society,” show a greater amount of survival and rearing among children born of women about that age than at any other; and recently Dr Arthur Mitchell has published a collection of cases of idiocy, with the respective ages of the mothers at the time of the idiots' births, and these also show a smaller proportion born of women about the age of 25 years than at greater and lesser ages.

1. The active ingredient of the poison of the Abor and Mishmi poisoned arrows is, in some of them, aconite, and, in others, croton oil. The former, apparently, is generally used by the Mishmis, and the latter by the Abors.

2. Although sufficient botanical materials have not been obtained to identify the species of aconite, the nature of the pharmacological action suggests that the species is one containing relatively more pseudo-aconitine than aconitine and, therefore, more resembling Aconitum ferox and A. heterophylloides than A. Napellus.

3. The arrow-poisons containing aconite were found to be much more lethal in warm-blooded animals than those containing croton; but the aconite-poisoned arrows that were examined carried usually too little aconite for a single arrow to produce death in man, even if the whole of the poison should be quickly absorbed.

4. The arrow-poisons containing croton, on the other hand, were found to be incapable of producing death in warm-blooded animals by the absorption of the poison into the circulation, and could do so only tardily by rendering the animal more susceptible to septicæmia, following inflammatory and even necrotic changes in the tissues into which the poison had been inserted.

5. While thus relatively inert in warm-blooded animals, and presumably, therefore, in man, the croton arrow-poison is extremely toxic in cold-blooded animals, being for them one of the most lethal of poisons, readily absorbable into the circulation and producing irritation and hæmorrhages in parts remote from the locality of insertion, and the latter, especially, in the alimentary canal. Remote effects may be produced even without any obvious evidence of local irritation in the place into which the poison has been inserted. These remarkable peculiarities in the action of the Government of India arrow-poison are reproduced by its ether extract and by the oil of Croton Tiglium.

6. Excepting failure to cause general action in warm-blooded animals, it is interesting to note that in many of their important effects, croton arrow-poisons as well as croton oil reproduce the effects of viperine venoms.

Dear Sir,

Some time ago I promised to send you the result of a series of observations, made to determine the situation of this place. Having, however, been much hurried of late, I am only able at present to transmit you the determination of the latitude, deduced from a series of observations of the sun's meridian zenith distances. With respect to the longitude, as soon as it is in my power, I will reduce some observations of occultations, and of the late solar eclipse, and send you the results.

The following experiments and observations have been undertaken with a view to discovering the means by which Tarsonemus is transmitted from one bee to another. It is obvious that a stage of the parasite exists outside the bee, and that there are also several possibilities (which may occur). One is the passage from bee to bee within the hive either directly or through the medium of frames or combs. Another, also within the hive, where the mites may in wandering upon the frames enter the cells and invade the body of the developing larvæ or pupæ, and in this way be present in the bee when it hatches. A third possibility is that whereby foraging infected bees may leave the mite upon flowers, vegetation generally, drinking grounds, or other situations, to be picked up later by other bees chancing to visit these. Crawling or dead bees may in a similar manner prove to be a source of infection through the contamination of the ground about the hive or of the actual hive itself.