(1) In the Cambridge Math. Journal, ii. p. 233, Sylvester showed how to eliminate x, y, z from the set of equations

His method consisted in deriving other three equations involving the variables x2, y2, z2, yz, zx, xy, and then eliminating these six variables from the six equations. The result obtained was

—a result which, on account of its complexity, it is impossible to rest satisfied with.

The manifest fact that when A = M = R = 0, the eliminant takes the form

and the further fact, that when a = m = r = 0, the equations become linear, with the eliminant

raise the presumption that Sylvester had not hit upon the simplest mode of performing the elimination.

The following investigation was undertaken in connection with a series of experiments on absorption at present being carried out in this laboratory.

In Gmelin's Handbook (Cavendish Edition, vol. 3, p. 240) it is stated, on Gay-Lussac's authority, that magnesia, when shaken up with a solution of iodine, absorbs the latter and assumes a reddishbrown colour. The production of this brown “hypoiodite of magnesium” is accompanied by the formation of small quantities of magnesium iodide and magnesium iodate, which pass into solution. When the brown compound is boiled with water, magnesium iodido and magnesia are formed, and when it is heated by itself, iodine is given off and magnesia remains.

The same compound is produced when magnesia is precipitated in presence of iodine, and its characteristic appearance makes it an excellent means of identifying a magnesium salt. The test is described as follows in the Notes on Reactions of Salts, used in the Practical Chemistry Class in the University of Edinburgh. “When a solution of iodine in potassium iodide is added to a magnesium salt, and a few drops of caustic potash are added to the mixture, a reddish-brown precipitate is produced; excess of caustic potash decolorises the precipitate.”

The state of the magnesia has apparently little effect on the production of the brown substance. It is formed indifferently with freshly precipitated magnesia, magnesia dried at the ordinary temperature, magnesia dried at 100°, and magnesia ignited over the blow-pipe.

Dr Munro maintained in this paper that, by the attainment of the erect posture and the consequent conversion of his limbs into hands and feet, man became homo sapiens, and inaugurated a new phase of existence, by means of which the manipulative organs became correlated with the progressive development of the brain. In the evolutionary career of man two stages were therefore to be recognised. First, that during which his physical transformation had been effected, so as to adapt him to bipedal locomotion; second, that during which his mental organisation had become a new governing force in the universe. The one, being readily effected in accordance with the laws of morphological adaptation, had a short duration. The other, an extremely slow process, consisted of small increments to his knowledge, acquired by repeated experiences of reasoning from causes to effects and from means to ends. The one was merely an adjustment of physical contrivances to physical ends, comparable to that by which the bird, the bat, and the whale had converted their limbs to their special purposes. The other had to be relegated to the mystic laboratory, where thought was converted into its material equivalent in the form of increased brain substance. The transition from the semierect to the erect posture could not, in point of duration, be at all paralleled with the ages during which this erect being had lived on the globe. It was also probable that this transformation took place in a limited area, so that the chances of finding the intermediary links of this stage were very small.

On 7th December last, I showed to the Society a method by which the skin might be stimulated by electric shocks from an induction coil corresponding in rhythm and intensity to the notes and chords of music given off by the phonograph. The pressures from the disk of the phonograph act on a variable resistance microphone transmitter. and the current passing through the latter also passes through the primary coil of an induction machine. From the secondary coil wires pass to two strips of platinum, immersed each in a beaker containing ·75 per cent. of salt solution. When the fingers are dipped into the salt solution, and the phonograph is set in motion, thrills are felt by the fingers corresponding to the intensity and rhythm of the music.

Deeming it of interest to have some data regarding this experiment, I have modified it in the way shown in the diagram, and the E. M. F. of the battery employed and the resistances of the various pieces of the apparatus were measured. Into the circuit of the secondary coil, f, a resistance box, g, and a telephone, t, were introduced; and the circuit was completed when the fingers were immersed in the salt solution h, i. When the phonograph was in action and gave out a well-known military march, the thrills could be felt in the fingers, and at the same time the telephone, which was fixed in a stand at the level of the ear, gave out the music.

There are two constituents of the cell which have, during the last few years, attracted a great deal of attention, namely, the nucleins and paranucleins.

The nucleins are essential constituents of the cell nucleus, appearing there in different forms of combination. The true nucleins possess at least three characters in common.

In the first place, they have a high percentage of phosphorus. Secondly, they offer great resistance to the action of peptic digestion.

Thirdly, they furnish nuclein bases on being subjected to the action of weak acids at moderate temperatures (about 100° C). They may be regarded as representing combinations between a rich phosphorus-holding acid—nucleic acid—and albumin. These combinations may either be of a firm or loose character, that is to say, in the case of the former it is difficult to split off the nucleic acid from the nuclein by the action of hot saturated baryta solutions, while in the latter the decomposition takes place easily.

It seems probable that there are four or more different nucleic acids, according as they give on decomposition certain definite nuclein bases. In other words, each nuclein base may have its corresponding nucleic acid from which it is derived. From this it follows that there may be also different nucleins corresponding to each nucleic acid. As examples, one may take thymus nuclein, which gives in decomposition with weak acids adenin and guanin, while, from the nuclein of the pancreas, only guanin has been obtained. This acid constituent of the nuclein possesses the property of precipitating proteids out of their solutions.

I have the honour to lay before the Royal Society a short account of some theoretical researches on the daily change of the temperature of the air, made by me in 1895, and published to a certain extent in the Nova Acta der kaiserlich Leopoldinisch-Carolinischen Academie der Naturforscher, Band lxvii., No. 2. Some more recent investigations on the subject, chiefly dealing with the physical point of view of the problem, have not yet been published, but seem to me of such importance as to justify me in presenting a brief sketch of them to the Society in this paper.

The history of the mathematical aspect of the problem can be traced back as far as the middle of the last century, when Lambert made the first attempt at investigating the relation between solar and terrestrial radiation and the temperature of the atmosphere. He was succeeded by a great number of famous mathematicians and meteorologists, whose labours, however, though they contributed considerably to our knowledge of certain parts of the question, have not been rewarded, it must be said, with any sufficient result; so that a satisfactory solution of the problem by means of theoretical investigations has been eventually considered impossible, at least in some quarters.

There can be no doubt that, in a problem like this, an immense number of different errors affect the principal conditions of the propagation of heat, which conditions are supposed to be the causes of the observed changes of the temperature.

It has for some time been the design of the Council of the Scottish Meteorological Society to establish on Ben Nevis a third observing station, supplementary to those at the summit and at Fort-William; and in the autumn of last year this design was partially realised. A site was not acquired so easily as might have been expected; but, after some disappointment, the Council, on the advice of Dr Buchan, who in person inspected the ground, fixed upon the best situation obtainable in the circumstances. Before the end of August the observer was in residence, but as the barometer did not arrive till the last day of the month, the complete record of observations extends only from the 1st to the 23rd September—23 days in all. The total number of readings taken was 186, an average of slightly more than 8 every 24 hours. Six of those observations were made at the hours of 9, 10, 12, 14, 18, and 21, while additional readings ran in a regular series between those fixed periods; thus accumulating a mass of facts for all times of the day and in various kinds of weather. The gaps are accounted for by the fact that, combined with my meteorological duties, were the functions of cook and maid-of-all-work, which occasionally rendered it impossible for me to leave the base of operations. Further, it was set down in my instructions that it was unnecessary to go out during continued bad weather, except at the statutory periods above mentioned; and that, in the event of extra readings being required during storms, a message would be sent from head-quarters.

Ethyl-sulphone-acetic acid, C2H5SO2 – CH2 – COOH, was prepared by Claesson's method. Fourteen grams of the potassium salt were dissolved in 20 c.c. of water, and the solution electrolysed for 30 minutes under following conditions:—Volts, 12; ampères, 3·4; temperature, 30°–35° C.

Carbonic anhydride and hydrogen were given off, and a white solid substance separated. This was collected on a filter and washed with cold water. The product, when dry, weighed 1·7 gram. It was purified by recrystallisation from chloroform.

It forms needles, melting at 136°–137°.

In the autumn of 1895 I received from Professor M'Ewen several samples of gastric juice marked “Chalmers' Gastrostomy.” The samples were obtained under most favourable circumstances, being drawn off through an aperture made in the abdomen and stomach of the patient; the juice was almost pure, and nearly free from suspended matter. My analysis was mainly directed towards ascertaining whether or not free hydrochloric acid was a constituent of the fluid, and the following were the reactions I employed:—

(1) With methyl violet the sample gave a distinct blue coloration, showing the presence of a mineral acid. Supposing the acidity to be due to lactic acid, this reaction would not take place, as on trying a dilute solution of lactic acid with this reagent no blue colour was produced.

(2) On treating an alcoholic solution of phloroglucin and vanillin (1 grm. of the former and ½ grm. of the latter made up with 50 cc.'s rectified spirits) with the gastric juice and evaporating slowly on a water-bath, a red colour was produced. Dilute hydrochloric acid gave the same result; lactic acid none.

In the opinion of some authorities the presence of free hydrochloric acid is attributed to the decomposition on evaporation of the chlorides present in the gastric juice by free lactic acid. With a view to ascertain whether this could happen or not, I evaporated solutions of chloride of calcium and lactic acid with both of the above reagents with a negative result.

§ 1. The thermometer described in the present communication is a constant volume one. Its advantages proceed mainly from a simple arrangement whereby the total pressure of the enclosed air, and hence its temperature, since these are proportional, is measured directly by the height of one column of mercury. The external atmospheric pressure is eliminated by the adjustment of an auxiliary reservoir of mercury.

§ 2. In fig. 1, A is the air-bulb, B is the pressure-gauge, which is an ordinary barometer tube with a vacuum above the mercury connected at its lower end to the bent stem of the air-bulb and also by a flexible tube to the reservoir C. D is the constant volume mark on the stem of the air-bulb. The part of the instrument which is filled with mercury when the air is at 0° C. is shaded in the diagram.

§ 3. To make an observation of temperature the mercury is adjusted to the mark D by raising or lowering the reservoir. The height of the column in B above the level of the mark D now gives the total pressure of the enclosed air. The air-bulb and pressure-gauge may be cut off entirely from external pressure by closing the stop-cock between the pressure-gauge and the reservoir. If the absolute temperature corresponding to any given value of the pressure be known, we can calculate that corresponding to any other pressure by simple proportion.

Thus, in order to graduate the thermometer in degrees of temperature, it is only necessary to find the total pressure for any one temperature; that of melting ice, which we may take as 273° on the absolute scale of temperature, is the most convenient.

A chief difficulty in the study of photo-chemical action lies in its variety, for it gives rise not merely to changes of a chemically simple kind, but also to the innumerable complex changes of plant-life and photography.

The difficulty is further increased by the extremely small proportion of material substance, chemically altered in a limited time by the action of those vibrations of the ether which we call light. To this, perhaps, more than to any other cause, must be attributed our continued inability to ascertain the true nature of photo-chemical changes so constantly observed as the decomposition of silver haloids by light.

Repeated attempts have been, made to connect particular chemical effects, such as oxidation and reduction, with greater or smaller wave-length; but all such attempts have ended in self-contradiction (1, 2).

Now, if there be any single characteristic associating all photochemical actions together, this characteristic must be present in the simplest as well as in the most complex cases, and the consideration of the former seems, therefore, most likely to lead to definite conclusions.

Perhaps the most striking and the most simple case that presents itself is the increased electric conductivity imparted to crystalline selenium on exposure to light.

Monkmann has shown that pure soluble sulphur also acquires increased conductivity when exposed to light, though to a less degree than selenium (3).

Ultra-violet light (4), and also the Röntgen rays, impart electric conductivity to the air.

In these eases the change is a temporary one.

In a paper read to the Society in May last, I treated specially the case in which the Hamiltonian cubic has all its roots real. In that paper I employed little beyond the well-known methods of Hamilton, but some of the results obtained seemed to indicate a novel and useful classification of the various forms of the Linear and Vector Function. This is the main object of the present communication.

It is known that we may always write

and that three terms of the sum on the right are sufficient, and in general more than is required, to express any linear and vector function. In fact, all necessary generality is secured by fixing, once for all, the values of α, β, γ, or of α1, β1, γ1, leaving the others arbitrary:—subject only to the condition that neither set is coplanar. Thus as a particular case we may write either

In either case we secure the nine independent scalar coefficients which are required for the expression of the most general homogeneous strain. But forms like these are relics of the early stage of quaternion development, and (as Hamilton expressly urged) they ought to be dispensed with as soon as possible.

2. A linear and vector function is completely determined if we know its effects on each of any system of three non-coplanar unit-vectors, say α, β, γ. If its cubic have three real roots, these vectors may, if we choose be taken as the directions which it leaves unaltered; if but one, we may take a corresponding system in the form

α, βcosa ± ιγsinα,

where ι is But it is preferable to keep the simpler form α, β, γ, with the understanding that β and γ may be bi-vectors, of the form just written.

§ 1. Many experimenters have investigated the electrical properties of flames and incandescent solids. The methods usually employed have been (1) to examine the electric conductivity of different parts of the flame; (2) to measure the difference of potential between platinum wires in different positions in the same flame; (3) to find the leakage of a charged conductor when placed near, or in view of, a flame or an incandescent solid; (4) to observe the leakage of a conductor, raised to a red or white heat, by an electric current, and electrically charged; and (5) to observe the production of electrification or diselectrification by a glowing wire, through which a current is passing, in neighbouring insulated conductors separated from it by different gases.

§ 2. This short communication divides itself into three separate inquiries: 〈1〉 to test by one of our electric filters the electric quality of the fumes from different flames and burnings (this method has not, we believe, been tried before); (2) to observe the difference of potential maintained between two wires of the same metal connected with a copper plate and a zinc plate when the fumes from different flames and burnings at different distances from the plates passed between them and round them; and 〈3〉 to observe the leakage between two parallel metal plates with any difference of electric potential when the fumes from flames and burnings were allowed to pass between them.

To approach the subject of osmotic pressure against an ideal impermeable membrane, consider first a vessel filled with any particular fluid divided into two parts, A and B, by an ideal surface, MM. Let a certain number of individual molecules of the fluid in A, any one of which we shall call D (the dissolved substance), be endowed with the property that they cannot cross the surface MM (the semi-permeable membrane); but let them continue to be in other respects exactly similar to every other molecule of the fluid in A and to all the molecules of the fluid in B, any one of which we shall call S (the solvent), each of which can freely cross the membrane. Suppose now the containing vessel and the dividing membrane all perfectly rigid. Let the apparatus be left to itself for so long time that no further change is perceptible in the progress towards final equilibrium of temperature and pressure. The pressures in A and B will be exactly the same as they would be with the same densities of the fluid if MM were perfectly impermeable, and all the molecules of the fluid were homogeneous in all qualities; and MM will be pressed on one side only, the side next A, with a force equal to the excess of the pressure in A above the pressure in B, and due solely to the impacts of D molecules striking it and rebounding from it.

The paper dealt with fifty-two species of butterflies and nineteen of beetles, the type-specimens of which had been discovered by the author in a collection purchased by the University of Edinburgh from M. Dufresne of Paris in the year 1819, and afterwards transferred to the Museum of Science and Art. In the case of the butterflies, the species referred to were described by Godart in the Encyclopédie Méthodique, while the beetles belonged to species described by Olivier in the same work, and also in his Histoire Naturelle des Insectes—Coléoptères, published about the same time. By the comparison of these original specimens with others in the Natural History Collections at the British Museum the author has been enabled to clear up many points in synonymy, etc., which have for nearly eighty years remained doubtful and obscure. The most important results of the investigations may be summarised as follows:—One of the beetles has been found by Mr Gahan, of the British Museum, to be the type of a new genus, which is characterised in the present paper, while the specimen upon which it is founded is probably unique; it has been found necessary to rename one species of butterfly and one beetle; errors in synonymy have been corrected in the case of nineteen species; and eight species hitherto wrongly placed have been referred to their proper genera.

1. The “dialytic” method of elimination, in the case of more than one variable, is not without its drawbacks, as most mathematicians know. The requisite derived equations are not always easily obtained, the difficulty being due as often to the existence of too many as of too few; and, when this has been got over, it not unfrequently happens that the order of the resulting determinant is alarmingly high, and unaccompanied by any hope of a successful guess as to the character of the extraneous factor. The discoverer's original paper affords sufficient testimony of this, and very little has been done since its appearance to put matters on a sounder footing. The most noteworthy improvement, due to Cayley, is more interesting theoretically than practically, his main object being the detection of the extraneous factor when there is an over-plus of equations. Nothing, indeed, will be found more conducive to an understanding of the limitations of the method than a careful comparison of the application of this process of Cayley's to the problem of eliminating x, y, z from the three ternary quadrics

with Sylvester's original treatment of the same problem, and the latter especially as commented on in footnotes by the author himself.

2. The want of definiteness in the mode of arriving at the exact number of derived equations is not without its compensating advantage, as it leaves that scope for the exercise of ingenuity which is half the charm of mathematical work.

1. Cayley commences his third paper on Skew Determinants (May, 1854) by recalling his development of them in terms of Pfaffians, and then goes on to say:—

“J'ai trouve recemment une formule analogue pour le developpement d'un déterminant gauche borde, tel que

Cette formule est:

and he explains that the expressions 12, 1234, etc., are Pfaffians, whose law of formation is—

12 = 12,

1234 = 12·34 + 13·42 + 14·23,

123456 = 12·34·56 + 13·45·62 + 14·56·23 + 15·62·34 + 16·23·45 + 12·35·64 + 13·46·25 +14·52·36 +15·63·42 + 16·24·53 + 12·36·45 + 13·42·56 + 14·53·62 + 15·64·23 + 16·25·34.

No proof is given, and the law of formation of the development itself is not explained.

1. In a paper “On the Existence of a Root of a Rational Integral Equation,” published in the Proc. Lond. Math. Soc., xxv. pp. 173–184, the author, Professor E. B. Elliott, says (p. 184) that it is unfortunate, for the simplicity of the argument of his paper, that a proof of a certain property of this eliminant, viz., that when two linear factors have been withdrawn from it, there is left a perfect square—“is one which direct algebraical methods have not yet supplied.”

In the course of the following year the want referred to received attention, a demonstration being given by Mr W. W. Taylor, in a paper entitled “Evolution of a certain Dialytic Determinant,” which was read before the same Society (see Proc. Lond. Math. Soc., xxvii. pp. 60–66).

I purpose here giving another demonstration, which I think has the merit of bringing out more clearly the character of the constitution of the eliminant, and in which is followed, at the same time, that direct and expeditious course most likely to be taken by a student familiar with the theory of determinants.

(1) If we think of the circulant C(a1, a2,…., an−1, an) as the result of the elimination of x from the equations

it is readily apparent that, corresponding to every rational factor of xn – 1, there must be a rational factor of the circulant. Thus the circulant of the 6th order

must have four rational factors, viz., the factor

corresponding to the solution x = 1 of the equation x6 = 1, the factor

corresponding to the solution x = − 1, and two other factors corresponding to the partial equations

x2+x+1 = 0, x2−x+1 = 0.

The following demonstration of the avidity for oxygen on the part of the living animal tissues was made incidentally while I was injecting the gelatine and soluble Berlin-blue mixture (Ranvier) into the blood-vessels of cats and rabbits in preparation of histological material. It is a fundamental fact that living cells must have oxygen continually supplied to them; the tissues crave oxygen, and will take it from any substance within their reach capable of yielding it (the inspiratory phase of the internal respiration). Normally the HbO2 yields it—the gas reaching the cells via the lymph. The left external jugular was chosen, and opened the instant the chloroformed animal's heart ceased to beat: thus, though the animal was “dead” as a whole, its tissues were alive for varying periods thereafter.

The fluid traversed heart and lungs before reaching the aorta and arterial circulation. On cutting up the organs I thought the injection must have “failed” in certain organs, for the kidneys and liver were markedly pale—pale green—and in some of the divisions of the portal vein the gelatine seemed uncoloured or “bleached” white. The lungs were of a deep blue; and such organs as brain and eye were blue, while there was no loss of colour where the fluid and the blood had mixed in the vessels (in a case in which the blood had not been first washed out of the vascular system). Here we had clearly a case of de-oxidation of the deep blue ferric ferrocyanide (Berlin or Prussian blue) to the pale green or almost colourless ferrous ferrocyanide, which, however, underwent re-oxidation on exposure to the air, for the blue colour returned progressively in from ten minutes to twenty-four hours after.