In the application of the Characteristic Function of Hamilton, or of any allied function, to the computation of a symmetrical optical system three steps are necessary. The performance of the system as a whole must be considered, and from this it appears that the aberrations may be derived from certain ‘aberration coefficients’ which occur in the expansion of an ‘aberration function’. In the second place, relations must be obtained giving the properties of the complete system in terms of the properties of the component systems, which, in general, will be single refracting surfaces; and finally, an evaluation of the coefficients must be made for the simple system—a single surface. For the first and third of these steps reference may be made elsewhere, and also for a general investigation of the second; a simple derivation is given, in the present note, of the necessary relations between a composite system and its components in the case where first order aberrations only are considered.

The view which the author has expressed regarding catalytic activity as due to frozen groups of atoms with strong specific external fields is further tested by a study of the stability of preparations of copper at the instant of formation, and during the subsequent life of the catalyst.

This investigation is to discuss some of the conditions which influence the absorption spectra of various “saturated” and “unsaturated” organic compounds. A condensed cadmium spark was used as the source of radiant energy; alcoholic solutions of the substances were employed, except for stearic acid, which was dissolved in ether. The vapours were examined in a quartz tube 200 mm. long, placed in a bath which could be heated to various temperatures.

The following is an attempt to account theoretically for the observed specific heat of water vapour. The usual theory of the vibrational specific heat together with the theory of the dissociation of the H2O molecule at high temperatures is found to account satisfactorily for the observed facts. It is shown, also, that at temperatures near 100° C. there is an effect due to polymerisation.

1. The brush discharges from a counting point have been examined with the microscope.

2. Magnetic fields have been shown not to affect counting under certain conditions.

3. The function of the series resistance has been further studied.

4. An approximate measurement has been made of the mechanical reaction on the electrodes.

5. An attempt has been made to detect hysteresis in the discharge, with negative results.

6. An effect of intense ionization has been noted.

In a recent paper, G. N. Lewis and D. F. Smith have discussed the problem of the velocity of chemical reactions, especially that of the only gas-reaction, the decomposition of N2O5, which had then been proved with sufficient accuracy to be kinetic unimolecular, i.e. unimolecular in the purely empiric sense of the word.

During the last year the homogeneous reactions of haemoglobin, and the heterogeneous reactions of the red blood corpuscle, have continued to furnish us with abundant opportunity for the application of our method for measuring the velocity of very rapid chemical reactions*. The recent types of apparatus, which we have devised, have not departed in essentials from the general principle previously employed, but a number of modifications and extensions have been introduced to meet with greater efficiency the special conditions which were imposed upon us. Since these developments would seem to be of much wider application than the special purposes for which they were in the first place designed, we think that it may be useful to workers in other fields to give a brief summary of their general features here. In the present paper we shall accordingly describe the methods we have adopted for 1. Very fast reactions. 2. Very slow reactions. 3. Very dilute solutions. 4. Small quantities of fluid. 5. Reactions following one another in rapid succession. 6. Reactions involving short-lived transient compounds. 7. The detection of concentration gradients in heterogeneous reacting solutions. Special apparatus has been devised to deal with each of these cases which will be briefly described in the following sections.

1. The relation

in which we suppose each of a, b, c, a2 − c2 to be different from zero, leads, from a value θ, to two values of φ, which we may denote by θ1 and θ−1. Each of these, put in place of θ, leads, beside the value θ of φ, to another value of φ, say, respectively, θ2 and θ−2. If θ2 or θ−2 be put for θ, the same relation leads to two values of φ, say, θ1 and θ3 or θ−1 and θ−3, respectively. And so on. It may happen that θn is the same as θ, in which case also θ−n is the same as θ; this we may express by saying that the relation is closed, or that there is closure, after n links. It is the object of the present note to express in reduced form, in terms of a, b, c, the condition that this may be so. Evidently, if n = pq, the condition of closure after n links is satisfied when the condition for closure after p links (or also after q links) is satisfied. But there is a condition for closure after n links which is not satisfied for any less number of links; this is the condition which we call the reduced or proper condition for closure after n links, and it is this which we seek to express.

The relation between the reaction velocity of catalytic dehydrogenation and the rate of passage of alcohol vapour over the catalyst has been found by experiment. At low rates of flow the concentration of the aldehyde and hydrogen in the effluent gas, and therefore over the catalyst, is comparable with that of the reactant. The whole of the surface is thus not covered with the reactant, and the adsorbed reaction products cause marked decrease in the reaction velocity.

When the concentration of the reaction products becomes sufficiently small, that is, when the rate of passage of the alcohol vapour exceeds some limit, the velocity is approximately independent of the rate of flow of reactant. With very high speeds the reaction velocity tends to fall again, but this is probably due to the difficulty of keeping the temperature of the catalyst constant.

The formula previously deduced connecting the partial pressure of the reactant in a mixture with the fractional reaction velocity has been verified for mixtures of alcohol vapour with water, acetone and benzene; and the composition of the adsorbed gas film with the fractional partial pressure of the gases for mixtures of carbon monoxide with oxygen on platinum.

The effect of mixture of vapours on the temperature coefficient of reaction is considered, and equations deduced to show the alteration produced. If the heat of desorption of the reactant is greater than that of the diluent, then the temperature coefficient is diminished, if they are equal there is no alteration, and if less the temperature coefficient is increased. Experiments show that water present in alcohol has little effect on the temperature coefficient of the reaction.

We recently had occasion to have a right-angled quartz prism made with the optic axis of the quartz perpendicular to one of the short sides of the right-angled triangle (Fig. 1). The prism has the property that it gives two images when light enters perpendicular to one of the faces and, after internal reflexion at the hypotenuse, passes out at right angles to the other face. It was at first sight rather difficult to see why this doubling of the image occurs, since the incidence on both faces of the prism is normal and quartz is a uniaxial crystal, there can be no double refraction occurring.

As a preliminary to an investigation of certain diffraction patterns I was led to consider, in some detail, the geometrical aberrations of a symmetrical optical system; and it appeared convenient then to classify the aberrations in orders according as they depend upon various powers of certain small quantities and to exhibit them as coefficients in the expansion of an ‘ Aberration Function.’ If aberrations of the first order only are considered, it becomes evident that one of them stands, in some sense, apart from the rest; I refer to the so-called ‘Petzval’ condition for flatness of field. It is of interest to notice that this condition was known to Coddington and to Airy before the time of Petzval—known at least as far as concerns systems of thin lenses. In the usual notation the condition is ΣΚ/μμ′ = 0; it is therefore independent of the positions of the object and pupil planes and in this respect it stands alone among the first order aberrations. But an increasing number of similar aberrations of higher orders will be found and it is of interest to examine these and to investigate their geometrical meaning. In the following note is given a proof of the Petzval condition, differing from that usually given and falling more into line with the general theory, and indicating also a general method of examining aberrations of this peculiar type.

The alkali metals, sodium, potassium and rubidium can be distilled easily in a good vacuum and obtained reasonably free from occluded gas in the following way. As potassium is now used for the absorption of mercury vapour and many experiments are being done on the others, an account of a convenient way of preparing pure specimens may be of some service to experimenters.

Expressions are found for the progress of slow selection in a Mendelian population where generations overlap. The changes are very similar to those which occur when generations are separate.

In the famous paper in which he developed his theory of radiation, Einstein investigated two problems. He investigated first the problem of what distribution of radiant energy in the steady state would be set up by an enclosed assembly of atoms in thermodynamic equilibrium radiating and absorbing according to the assumptions of his theory; he showed that the distribution so set up was given by Planck's law. He investigated secondly the velocities set up amongst the atoms in consequence of the random variations in direction of the emissions and absorptions; provided that emissions are assumed to be directed, so giving rise to recoil momentum, he showed that the mean square velocity-component in a given direction, , is equal to its equipartition value, given by .