This third and last part of the paper is concerned with the interpretation of the Schläfli symbol {k1, k2, …, km−1} when the k's are unrestricted. It is shown that, whenever the k's are integers (greater than 2), the symbol represents a polytope in a generalized space having time-like as well as space-like dimensions.

The object of this paper is to obtain the general solution to the self-adjoint partial differential equation in n dimensions

where pij, q and ρ and bounded, continuous functions of (x1,…, xn) in a domain D and on its boundary, and where ∑pijXiXj≥0 for all (x1,…, xn) of D and all X1,…, Xn. The domain D is an n-dimensional domain and may be either the whole or part of a Riemann surface space of n dimensions. Its boundary is to consist of any number, zero, finite or enumerable, of continuous continua of n − 1 dimensions. These terms will be explained in paragraph II. The solution u = u(x1,…, xn; t) will be valid for (x1,…, xn) in D and t ≥ 0, and will satisfy boundary conditions of the type or similar, these conditions becoming identical at any part of the boundary of D that lies at infinity.

1. In a paper read to this Society Mr F. P. White has derived a remarkable chain of theorems in [n] which, for n = 2, becomes Clifford's chain. White's method is so elegant that, as he has restricted himself to curves going through two given points, it may be permissible to shew that his line of argument applies equally to curves going through three or more given points, thus producing other chains of theorems analogous to Clifford's. For this purpose it is necessary to consider the successive projections, not of a simplex, but of a figure of n, n − 1, n − 2,…vertices in [n].

The investigations which follow were originally suggested by the now classical problem of Cayley, the determination of the condition that seven lines in space, of which no two intersect, should lie on a quartic surface. This problem suggests the consideration of the linear system of quartic surfaces which pass through six given lines, and this, essentially, is the basis of all that follows.

1. In this paper I discuss the expression of the general form F of order n, homogeneous in r + 1 variables z0, z1,…, zl,…, zr, as the sum of the nth powers of h + H linear forms in these variables. I take h (> 0) of these linear forms to be undetermined, namely the forms

whose coefficients are undetermined; and I take the remaining H(≥0) linear forms to be assigned, namely the forms

whose coefficients are assigned.

1. If a class consists of n members of which r have the property φ, and we wish to estimate r by the process of sampling, the distribution of probability among values of r, given the composition of the sample, necessarily depends to some extent on our previous knowledge. This is expressed by saying that the prior probability of a particular value of r is f(r). According to Laplace's theory f(r) is constant. It is obvious that there are cases where the previous knowledge is such as to ensure that f(r) is not constant; but the question remains whether such cases should be considered the exception or the rule.

1. In a previous note the author has examined the systems of tangent planes to a degenerate surface in S3 consisting of n planes, regarded as the limit of a general surface of the same order. It is well known that a pair of space curves which are the limiting form of a non-degenerate curve must have a certain number of intersections; hence, if a surface in higher space degenerates into a number of surfaces, these must intersect in curves of various orders. In the present paper we consider the nature of the envelope to a surface consisting of two general surfaces of S4 having in common a single curve of general character, the degenerate surface being regarded as the limit of some surface of general type. The same conclusions hold for a similar degenerate surface in Sr (r>4).

The following paper arises from a remark in a recent paper by Professor Baker. In that paper he gives a simple rule, under which a rational surface has a multiple line, expressed in terms of the system of plane curves which represent the prime sections of the surface. The rule is that, if one system of representing curves is given by an equation of the form

the surface being given, in space (x0, x1,…, xr), by the equations

then the surface contains the line

corresponding to the curve φ = 0; and if the curve φ = 0 has genus q, this line is of multiplicity q + 1.

1. It is known that, in [3], a ruled surface of order n and genus p has in general a double curve of order ½ (n − 1) (n − 2) − p and genus ½ (n − 5) (n + 2p − 2) + 1, 2(n + 2p − 2) torsal generators, 2(n − 2)(n − 3) − 2(n − 6)p generators which touch the double curve, and triple points.

The effect of a monomolecular film on the reflexion of light from a plane surface is considered. The film is treated as a two-dimensional assembly of scattering centres, and the possibility of molecular orientation is allowed for. Experimental results, when available, should indicate whether the scattering properties of groups of molecules in the “two-dimensional” state differ from those of molecules in bulk.

According to the Frenkel theory of adsorption, the first molecules striking a surface move about on it like a two-dimensional gas before finally being adsorbed. This surface motion has been demonstrated by Estermann and by Cockcroft in the case of vaporised cadmium.

The same perturbation method is found to be valid for a gentle mutual potential between free atom and vibrator as for an infinitely steep mutual potential, provided that the reduced mass of the system is sufficiently small.

A simplified method is discussed for the calculation of transition probabilities. This method is used to calculate the effect of a van der Waals' attractive field.

The partition function F(θ) is of fundamental importance in the theory of the specific heat of gases. Once it is known, the rotational specific heat of a perfect gas is given by

where R is the gram-molecular gas constant, and θ bears the relation

to the absolute temperature T, k being Boltzmann's constant.

The phenomena which limit saturation in a high pressure ionisation chamber are examined with particular reference to the rôle played by preferential recombination. The complexity of the phenomena is discussed in detail in order to bring out the scope and limitations of the simple theory of preferential recombination and the conditions which a more complete theory would have to satisfy. It is pointed out that in a number of recent papers on the subject fundamental aspects of the problem have been overlooked, and the significance of these omissions is discussed.

Using a thin-walled electroscope, absorption curves in paper and aluminium were obtained for the β-rays of actinium (B + C) and actinium C″. From these curves the range of the β-rays of actinium B was found to be 0·08 ± ·02 gm./cm.2 The energy required for this range is 300,000 ± 50,000 volts, which must be the end-point of the continuous spectrum.

On account of the overlapping of the β-ray spectra of actinium B and actinium C″ the range method can never yield an accurate value of the end-point of the former. Magnetic analysis appears to be the best means of finding this accurately, but the weakness and rapid decay of sources are formidable difficulties.

The theory of the evaporation of adsorbed atoms and ions of the alkali metals from a hot metal surface has been developed from somewhat different viewpoints by Langmuir and Kingdon and by Becker. These two treatments, which yield essentially the same results, are applicable only when equilibrium conditions obtain on the surface, so that the surface concentration is not changing. It is the purpose of the present note to extend Becker's treatment to take account of the rate of attainment of equilibrium on the surface, in order to lead to an interpretation of some experiments on the rate of evaporation of the alkali metals adsorbed on hot tungsten.