The response by a plant protoplast to an inducing or to an inhibitory stimulus is preceded by a period of non-responsiveness, and the withdrawal of the stimulus is succeeded by a period of continued response or inhibition as the case may be. During these intervals, the familiar latent period and period of after-effect, or temps de memoir of Massart, the protoplast manifests the property of functional inertia (Harris, Brit. Med. Assoc., 1900), and in the case of the inducing stimulus the two phases follow in the order of (1) the Anabolic and (2) the Katabolic, while in inhibition the order is reversed. This property is exhibited by the protoplasm of growing and of adult organs, as well as by that of isolated and excised organs; further, it varies in amount and appears under different aspects.

§ 1. The object of this communication is to partially realise the hope expressed at the end of my paper of July 1 and July 15, 1889, on the “Molecular Constitution of Matter”:—“The mathematical investigation must be deferred for a future communication, when I hope to give it with some further developments.” The italics are of present date.

Following the ideas and principles suggested in §§ 14–20 of that paper (referred to henceforth for brevity as M. C. M.), let us first find the work required to separate all the atoms of a homogeneous assemblage of a great number n of molecules to infinite distances from one another. Each molecule may be a single atom, or it may be a group of i atoms (similar to one another or dissimilar, as the case may be) which makes the whole assemblage a group of i assemblages, each of n single atoms.

The above question having been set before me by my friend Prof. P. G. Tait, and in such a guise that its practical solution, if amounting to anything sensible, might have astronomical applications, I set myself to examine it with the highest dispersion power in my possession, viz., a fine Rutherfurd Diffraction grating, of 17,296 lines to the inch, ruled over a surface 1·6 inch square; a telescope 52 inches long, with magnifying powers from 20 to 50; and a collimator 32 inches long, armed with a very substantial slit-apparatus by Mr Adam Hilger, and some other fittings.

When thio-urea is treated with suitable oxidising agents in presence of acids, salts are formed corresponding to the general formula (CSN2H4)2X2:—

Of these salts the di-nitrate is very sparingly soluble, and is precipitated on the addition of nitric acid or a nitrate to solutions of the other salts. The salts, as a class, are not very stable, and their solutions decompose, especially on warming, with formation of sulphur, thio-urea, cyanamide, and free acid. A corresponding decomposition results immediately on the addition of alkali, and this constitutes a very characteristic reaction for these salts.

As is well known, the usual form of the Addition-Theorem for Elliptic Functions of several arguments expresses these functions as the quotient of two determinants. When two or more arguments become equal, both numerator and denominator of this quotient vanish, and in seeking to remove the common vanishing factor, Cayley, in his paper “Note sur l'addition des fonctions elliptiques,” in connection with the cases of three and four arguments, brought to light some identities connecting certain alternants. Cayley gave these identities without proof, saying, “Je n'ai pas encore trouvé la loi générale de ces équations.”

The special form of determinant to which we have now come is connected with a problem in coordinate geometry—the problem of transformation from one set of rectangular axes to another set having the same origin. The actual appearance of determinants in any of the attempts to solve the geometrical problem did not take place until comparatively late in its history, probably because the connection between the two subjects was less patent than in other cases, the problem when transformed into algebraical language being not a mere matter of elimination of unknowns from a set of linear equations. The earlier portion of the history of orthogonal substitution, although of considerable interest, is thus not sufficiently germane to our subject to warrant detailed treatment of it. For those interested in this earlier portion it will suffice to give the following chronologically arranged list of papers from 1770 to 1840:—

In examining stained blood films we have frequently observed necrobiotic changes in the white cells. These changes have been referred to by different authors. Cabot (1) writes, “frequently in leukæmia and occasionally in other conditions one sees leucocytes apparently moribund. That they are not always artifacts is shown by the fact that in normal blood they do not appear when treated by the same technique, that reveals them in the blood and hardened clot of leukæmic cases, as well as by the fact that Botkin and others have produced similar appearances by keeping the leucocytes a few days in an aseptic state.”

Rhombic and monoclinic sulphur are two physical states of the element with a transition point at 96°. The relation of amorphous sulphur to liquid sulphur is entirely different. The former is produced by heating liquid sulphur, and is found in increasing proportion as the temperature is raised. At 448° it reaches about forty per cent. As the temperature falls, the proportion of amorphous sulphur recedes, although sudden cooling arrests the regression, and furnishes solid specimens containing the larger proportions proper to higher temperatures. The case seems to be one of chemical equilibrium, S (liquid) ⇌ S (amorphous), in which the action, as written, is endothermal.

The diagram outlines the Trisector as in the position in which the angle B A L, which may be any angle of less than 225°, is by its aid divided into five equal angles.

That instrument was described to this Society in a communication submitted to them on the 4th of November last. It consists of two pieces, A I F D and BHG. Those pieces are conjoined, as the blades of a pair of scissors are, by a small cylindrical pin inserted in a small cylindrical hole at C in each of them, which it fits, and round which they may turn.

In a note on “the hydrolysis of thallic sulphate” which I communicated to the Society some time ago, I commented on theconflicting nature of the statements, made by various investigators, as to the exact composition and nature of thallic sulphate. As the subject appeared of some interest, I subsequently commenced an investigation of the salt and of the double salts derived from it. The results so far obtained are in some respects rather striking, but the investigation is not yet completed, and my reason for now publishing a general statement of these results is, that apparently others besides myself are working in the same field. A paper has just been published by James Locke, in which the author describes a caesium thallic sulphate which he obtained while endeavouring to prepare cæsium thallic alum. No alum could be obtained, the most hydrated salt corresponding to the formula CsTl(SO4)2, 3H2O.

A communication was made to this Society on February 3, 1902, on ‘Magnetic Shielding in ollow Iron Cylinders.’ At an early stage of this investigation, and when dealing with more magnetising forces than that due to the transverse field acting upon the shield, the subject of cross magnetisation necessarily came to the front. This paper deals, First, with the superposition of two magnetising forces at right angles to each other, and the coordination of the two components of the resultant magnetic induction, under the various conditions of field superposition; and Second, with the magnetic œolotropy of demagnetised iron. The same hollow iron cylinders were used as in the shielding experiments.

1. A quaternion r = Sr + Vr denotes the sum of a scalar and a vector, the former being an essentially undirected quantity. In many cases, however, and specially in the theory of screws, we have to deal with two co-directed quantities. In the usual notation, the components of a translation, λ, which are parallel to, and perpendicular to, a rotation, μ are represented respectively by the first and second terms in the identity λ = (Sλμ−1 + Vλμ−1)μ. The axis of the corresponding screw of pitch Sλμ−1 has the direction of μ., and Vλμ−1 is the perpendicular upon it from the origin. A certain advantage in point of unity would arise from taking Sr and TVr to represent respectively the magnitudes of the translation and the rotation in a screw whose axis passes through the origin, and has the direction of Vr. When such a use is made of a quaternion, it is necessary to attach a special symbol. Thus, Mr may be taken to denote the motor Sr·UVr + TVr·UVr whose axis passes through the origin and whose pitch is Sr/TVr.

The following observations are based chiefly upon the results of an exhaustive survey of the specimens in the Scottish Mineral Collection in the Edinburgh Museum of Science and Art, which were made while I was studying the material for completing The Mineralogy of Scotland. The survey has been continued since the publication of that work, in connection with the revision of the arrangement of the specimens in question. A large number of facts of interest have come to light in the course of this work. Furthermore, the forms of several hundreds of crystals in the Collection have now been determined, and freehand drawings of most of these crystals have been made, and placed alongside of the specimens to which they refer. It is proposed from time to time to lay some of the more interesting of the results arising from this work before this Society, especially as little or none of them have hitherto been published, and as the crystals, which present considerable interest, do not appear to have been previously figured.

While the investigation of the freezing-point of sulphur, described in these Proceedings (vol. xxiv. p. 299), was in progress, some experiments on the maximum quantities of amorphous sulphur, formed at various temperatures from 130° to 448°, were being made. The results showed remarkable irregularities. It was also found that the addition of foreign bodies, which are for the most part without chemical action on sulphur, greatly magnified these disturbances, and led to the discovery of their source. The following is a condensed statement of the principal facts.

The following quaternion notes are interesting as the last piece of mathematical work written down by Professor Tait. During his last severe illness he referred several times to the importance of working out still further the theory of the linear vector function; but he had not been able to put pen to paper for months previous to July 2, 1901, just two days before his death. On that date he wrote down the notes now printed, probably at intervals throughout the day. They would serve, in all likelihood, chiefly as nuclei round which his thoughts would circle; and they will now serve to indicate in some measure the directions in which his thoughts were moving. The foolscap sheet on which the notes were written was handed to his eldest son, with the request that it should be preserved, as he believed it contained the germ of an important advance. A facsimile of this sheet, on a reduced scale, is reproduced in the accompanying Plate.

Several years ago, Dr Veley and myself were engaged in making a prolonged series of measurements of the electric conductivities of nitric acid solutions of various strengths. During the course of the work we were repeatedly impressed by the fact that a very slight variation in the strength of a solution produced a very marked change in the value of the electric resistance, as measured in a somewhat high resistance cell.

The following research was undertaken for the purpose of determining the freezing points of solutions under conditions which would involve a different systematic error from that encountered when the ordinary method of procedure is followed; and, secondly, for the purpose of obtaining ionisation values for electrolytic solutions by the cryoscopic method which should be made by compensation as far as possible independent of any systematic error in determining the freezing points.

There is a short paper in a recent volume of the Comptes Rendus of the French Academy of Sciences which deserves notice if only in order that the attention of the author and others may be drawn to previous work on the same subject and to more effective methods of treatment.

(1) From a general theorem, known since 1855, and perhaps earlier, regarding the reciprocal of the series

it follows that

where β0 = 1 and

The one constant factor in the production of cyanosis is diminution of oxygenation. This may be the result of respiratory affections which hinder the access of air or diminish the area of the aerating surface, or it may, on the other hand, be due to circulatory disorders lessening the amount of the blood flowing to the lungs. These latter affections furnish the most conspicuous examples of cyanosis, and in them the symptom is entirely produced by hindrance to the blood flow. Such is in reality the original conception advanced by Morgagni and supported by numerous observers. The other view, apparently suggested by Corvisart, that the condition is due to admixture of arterial and venous blood, is absolutely untenable, as in the vast majority of cases there is no possibility of such intermingling. In general terms the blood in cyanosis may be said to be of high specific gravity—from 1070 to 1080. The amount of hæmoglobin rises considerably—often reaching above 100 per cent. The erythrocytes increase in number so as frequently to exceed 7,500,000 per cubic millimetre, while the leucocytes, although not so commonly altered in number, often reach such a figure as 12,000 per cubic millimetre. The increase of these constituents of the blood was first observed by Toeniessen. Malassez showed that there are more red corpuscles in the blood of the superficial than of the deeper parts of the body. Toeniessen and Penzoldt suggested that this might be due to loss of fluid from the surface, while the interior receives a constant supply of fluid from the alimentary tract.