The question has frequently been raised, whether the laws of ordinary Geometrical Optics, especially with regard to the reflexion and refraction at the confines of two media, are equally applicable when the transition is accomplished in a gradual manner. The problem has an important bearing on physical phenomena of a widely varying character, such as the propagation of sound and light in the atmosphere, with occurrence of mirage or zones of silence, the transmission of seismic disturbances through the earth, tidal waves in a canal of varying cross-section, and others.

Perhaps the most peculiar fact of observation yet recorded about the wing disposition of soaring birds is the apparent large negative angle of incidence shown by the wings of vultures when in fast horizontal flight.

§ 1. The groundwork of the present paper is a general transformation of Maxwell's equations. As relativist speculation had its origin in a simple transformation, it was to be expected that something more would result when the general transformation was explored: the determining influence has proved stronger than I had anticipated.

The only really useful practical method for solving numerical algebraic equations of higher orders, possessing complex roots, is that devised by C. H. Graeffe early in the nineteenth century. When an equation with real coefficients has only one or two pairs of complex roots, the Graeffe process leads to the evaluation of these roots without great labour. If, however, the equation has a number of pairs of complex roots there is considerable difficulty in completing the solution: the moduli of the roots are found easily, but the evaluation of the arguments often leads to long and wearisome calculations. The best method that has yet been suggested for overcoming this difficulty is that by C. Runge (Praxis der Gleichungen, Sammlung Schubert). It consists in making a change in the origin of the Argand diagram by shifting it to some other point on the real axis of the original Argand plane. The new moduli and the old moduli of the complex roots can then be used as bipolar coordinates for deducing the complex roots completely: this also checks the real roots.

In a paper on random flight Lord Rayleigh proved the following result: A number is formed by adding together n numbers each of which is equally likely to have any value from − a to + a. Then, if f (n, s) ds is the probability that the number so formed lies between s and s + ds, and if n is sufficiently great,

This result may be stated as follows: A point moves discontinuously in a straight line. For a time τ it has a constant velocity. During the next time-interval τ it again has a constant velocity, and so on. Then if each of these velocities is equally likely to have any value from − v to + v, the probability that in the time nτ, the point moves a distance lying between s and s + ds is f (n, s) ds, with vτ written for a.

This problem is in general treated in connection with the division of the periods of the elliptic functions. It is the object of the present note to shew, from a purely algebraical point of view, that the condition of closing of the polygon depends solely on a difference equation of the form

The following paper aims at a more general treatment than has hitherto been given, of the integral expansions of arbitrary functions, from the point of view of integral equation theory.

Stable colloidal sols are always charged, and disperse systems in water appear in most cases to acquire a constant electrokinetic potential of 70 m.v. When the electrokinetic potential falls to 30 m.v. coagulation commences and the rate of coagulation is, as Hardy first pointed out, most rapid at the isoelectric point. Thus the question on what the stability of a colloidal system rests must ultimately be referred to the magnitude of the electrokinetic potential and the methods by which this is increased or decreased in solution.

§ 1. Most of the methods which have been devised to give an equation of state of more generality than that of Van der Waals differ from the original method used by him in that they refer only to the uniform conditions in the interior of the gas while his had special reference to the conditions at the boundary. In fact, one of the corrections to the perfect gas law introduced by him is due entirely to the existence of a boundary field of force. The question arises as to what physical interpretation is to be given to the more general equation. The methods previously employed leave the interpretation obscure. In this note two new methods of obtaining the equation of state are given, one applicable to the interior of a gas, the other to the boundary. These methods seem to have certain advantages over those previously given in that they lend themselves to a very simple physical interpretation. While the pressure at the boundary of a gas is the same as that in the interior, the word pressure has a different meaning in the two cases. At the boundary, it is due entirely to the motion of the molecules, whereas in the interior part only is due to the motion of the molecules; this part is the same whatever the nature of the molecules and is in fact given by the perfect gas law. The remaining part is due to the stress set up by the existence of intermolecular fields and, although at any given point it is a fluctuating function of the time, it has everywhere within the gas the same statistical mean value. In this paper, it is referred to as the statical pressure to distinguish it from the more usually understood dynamical or momentum pressure.

If a gas, each of whose molecules is capable of dissociating into two similar molecules, be kept at a constant temperature, it would attain an equilibrium state in which the rate at which double molecules dissociate is equal to the rate at which they are formed by recombination of single molecules, there being a different equilibrium state for each different temperature. If, now, the gas be subjected to a temperature gradient, the concentrations necessary for equilibrium would be different at different points, so there would be diffusion between the two kinds of molecules, a steady state being attained when the rate of diffusion of double molecules into any region is equal to their excess rate of decomposition over their rate of formation within that region.

§ 1. The general quaternary cubic

can be expressed in various interesting canonical forms involving suitably chosen linear forms Xi, Yi, Ai. Thus, referred to the pentahedron. X1X2X3X4X5 of Sylvester the cubic becomes the sum of five cubes

with its Hessian in the form

where the coefficients ai may if necessary be taken as equal to unity and the five linear forms Xi each contain four independent parameters, making a total of twenty parameters which is the number of coefficients aijk in the given cubic C3

1. In a paper “Beiträge zur Inversionsgeometric,” which will appear in the forthcoming volume of the Science Reports of the Tôhoku Imperial University, I have treated the problem of determining the necessary and sufficient condition in order that two plane curves which are given by natural equations should be transformable into each other by inversion. In connection with that paper I propose here to treat the analogous problem for Laguerre transformations:

Having given two plane curves, by natural equations, in which the functional relations are all supposed to be analytic, it is required to determine the necessary and sufficient condition in order that the two plane curves should be transformable into each other by a Laguerre transformation.

The theorem, due to Miquel, that the foci of the five parabolas which touch fours of five straight lines lie on a circle, when generalised projectively and dualised becomes the theorem: If six arbitrary points 1, 2, 3, 4, 5, 6 be taken and the five conics passing respectively through the five points obtained by omitting in turn 1, 2, 3, 4, 5, then there exists a conic touching two arbitrary lines through the point 6 and triangularly inscribed to these five conics. It appears, however, that the relation is symmetrical and that the conic obtained is also triangularly inscribed to the conic passing through the points 1, 2, 3, 4, 5. If the condition of touching the two arbitrary straight lines through the point 6 be omitted, we have a doubly infinite system of conics triangularly inscribed to the six conics passing through fives of six points. It does not immediately appear how this family of conics depends upon the two parameters involved, and the following direct analytical investigation of the general symmetrical figure was undertaken with a view to deciding this point.

The instrument to be described is a modification of an electrostatic oscillograph originally invented by Prof. Taylor Jones. In the original instrument a thin phosphor-bronze wire is stretched parallel to a flat metal plate towards which it is attracted when a difference of potential is established between them. A small mirror is fastened by one edge to the middle of the wire, and the other edge of the mirror is pivoted by bringing up to it a flat cork against which it presses. The motion of the wire is recorded by reflecting a beam of light from the mirror on to a rotating drum.

Radiologists have often had cause to note certain apparent anomalies in the behaviour of their apparatus. It is quite well known that different X-ray tubes, excited by different kinds of high tension apparatus, yield X-radiation of markedly varying quality and quantity even under conditions which, as measured by spark gap and milliammeter (the usual measuring instruments of the radiologist), are apparently identical. These anomalies seemed to offer an interesting field for investigation, and one which might not be without value on the practical side.

The theorem that, if four arbitrary lines be taken in a plane, the four circles about the triangles formed by threes of these lines, meet in a point, can be generalised to space of any even number of dimensions, as was recognised by Mr J. H. Grace in 1897. In the plane case the centres of the four circles lie on another circle passing through the point of concurrence of these four; it has been sought to prove that the centres of the n + 2 spheres, similarly arising in space of an even number, n, of dimensions, also lie on a sphere†.