For a strut under thrust when buckling is resisted by a force proportional to the displacement, it is shown that for both a clamped and a pinned strut there is stability for any length when the thrust is less than a certain critical value; and the relation is found between the length and the first value of the thrust above this that will give buckling. A notion of the wave-length and number of nodes after buckling is obtained.

The results are applied to the theory of the formation of mountains by horizontal compression, and it is shown that the crust of the earth would be able to transmit, without buckling, stresses right up to the breaking stress across regions of continental extent, unless the depth down to the level of no strain were less than 16 metres.

An attempt is made to employ the results also to consider the stability of solid surface films under compression, and it is seen that other factors, making for stability, must enter besides rigidity and gravity. This is supplied by surface tension, and it then appears that collapse due to a weakness in the film must precede buckling.

A rough calculation is given of the frequency of vibration of an atom in a solid film.

The figure in three dimensions consisting of two sets of three non-intersecting lines, such that each line of one set intersects each line of the other set, will here be called a “net.” All nets are projectively equivalent to each other: and there are just 72 projective transformations that change one net into another.

There are in the Fifth Book of Euclid's Elements twelve propositions, the object of which is to prove the equality of two ratios.

The theorem is attributed to Wallace that, in a euclidean plane, the circumcircles of the triangles determined by four lines, of general position, meet at a point. It is further known that, in euclidean space of n dimensions, the circumhyperspheres of the simplices determined by n + 2 flats, of general position, meet at a point, if and only if n be even.

Any two real numbers p and q will be called conjugate if

Expressions are found for the changes caused by slow selection in populations whose characters are determined by incompletely dominant, multiple, or polyploid factors, and for the equilibria attained in certain of these cases.

1. During the last sixty years the principal questions presented by the higher singularities of plane algebraic curves have been completely solved, and definite results obtained. The two most successful lines of research have been by expansions and quadratic transformation. By each method it has been shown that a higher singularity may be looked upon as containing concealed or “latent” multiple points or lines in addition to those immediately recognized; and from each, with the help of small quantities, has been constructed a topological explanation of these latent multiple elements, which are accounted for as situated in the immediate vicinity of the point and line base of the singularity. Further, by each method it has been proved that as regards the numerical relations known as Plvicker's equations a singularity produces the same effect as a definite number of nodes, cusps, bitangents, and stationary tangents.

An account is given here of the measurements of the mass absorption coefficients of iron, nickel, copper and aluminium over a wave-length range 0·705 Å.U. to 1.932 Å.U.

The problem of the scattering of radiation by a free electron has been treated by the author on the basis of Heisenberg's matrix mechanics, which was first modified to be in agreement with the principle of relativity. The main point of this modification is that, whereas in the non-relativity theory one deals with matrices whose elements vary with the time according to the law eiwt, in the relativity theory the elements of the matrices must vary according to the law eiwt′ where t′ = t − (l1x1 + l2x2 + l3x3)/c if they are to determine correctly the radiation emitted in the direction specified by the direction cosines (l1, l2, l3), x1x2 and x3 being the coordinates of the electron at the time t. These matrices were obtained by writing the Hamiltonian equation of the system in the form

where W′ is a variable canonically conjugate to t′ and H′ commutes with t′, and then using H′ as an ordinary Hamiltonian function of a dynamical, system that has W′ for its energy and t′ for its time variable.

A century ago geodetic and gravitational universal surveys were mainly concerned with determining the effective (gravitational) ellipticity of the Earth, after due allowance had been made for local anomalies, with especial view to the exact purposes of physical astronomy. One of the chief of these anomalies was exhibited by a remark of Airy, after scrutiny of the available data in his treatise (1830) on “Figure of the Earth” in the Encyclopedia metropolitana, that the observations show gravity to be abnormally in excess on island stations. It appeared, for instance, that this cause might make the mass of the Moon uncertain up to 2 per cent. A very refined explanation of this anomaly of island stations (which will be seen presently to be only partially effective) was offered by Sir George Stokes, from whom this last remark is quoted, in the course of a memoir, fundamental for theoretical geodesy, in which he demonstrated that no outside survey could lead to any certain knowledge of the distribution of mass inside the Earth, even in its outer crust, except as a matter of probability when backed up by geological knowledge.

It is a familiar fact that if two quadric forms, in (n + 1) homogeneous variables, be each expressible as a sum of squares, of (n + 1) independent linear functions of the variables, then they are polar reciprocals of one another, in regard to any one of 2n quadrics. The question arises whether this is true for any two quadrics. Segre states that this is an unsettled question. A solution is given, however, by Terracini for any two non-degenerate quadrics, supposed to have been reduced to the Weierstrass canonical form. The present note has the purpose of pointing out that a solution is derivable from a remark made by Frobenius; this requires a knowledge of the roots of the equation satisfied by the matrix of the two quadrics.

In his remarkable memoir ‘On certain arithmetical functions’* Ramanujan considers, among other functions of much interest, the. function τ(n) defined by

This function is important in the theory of the representation of a number as a sum of 24 squares. In fact

where r24 (n) is the number of representations;

where σs(n) is the sum of the 8th powers of the divisors of n, and the sum of those of its odd divisors; and

A Fourier integral will be described as of finite type if the range of integration of the integrals by means of which the coefficients in the Fourier integral are defined is a finite interval instead of the usual (−∞ ∞). Thus is of finite type (p, q), such that where ƒ(x) is the generating function of the series.

1. Polarisation measurements are described on the Hg lines λ 4358, 5461, 5770, 5791 emitted from a low-pressure electron-maintained arc to which a longitudinal field of up to 3000 gauss may be applied.

2. Two types of polarisation effect have been found. The first is typified by the line λ 5461. It is a polarisation of the light with the electric vector perpendicular to the lines of force. Its maximum value is 9% but it decreases to zero (a) with decreasing field strength and (b) with decreasing current in the tube. It is suggested that this effect is due to a self-reversal of λ 5461 owing to the metastability of the 2 3p2 state and that the polarisation may be caused through a slight non-uniformity of the magnetic field.

The second type of effect is found for λ5770 and 5791. The polarisation is in a plane parallel to the discharge, and is found in a zero magnetic field. This effect is further discussed elsewhere. The line λ 4358 is unpolarised under all conditions tried.

3. Neither of these effects implies a polarisation of the total light emitted directly from atoms in the Transverse Zeeman Effect when the atoms are excited under isotropic conditions and the experiments give definite evidence against the existence of such a polarisation effect.

The theorem which we prove here seems obvious enough when stated, but it appears to have been overlooked by the numerous writers who have discussed the subject, and the proof is less immediate than might be expected.

The following gives a brief summary of a paper, which it is hoped to publish in extenso later. It is a contribution to the theory of ferromagnetic crystals, and contains in particular a theoretical explanation of Webster's experimental results.

Thomas's differential equation for the average field inside a heavy atom is analogous to Emden's equation for the polytropic equilibrium of a star. Emden's result that the total gravitational potential energy of a star is calculable once the differential equation has been solved is adapted to give the total electrostatic energy, and hence the total energy of binding, of an atom built on Thomas's model. This should be equal to the sum of the successive ionisation potentials. The total energy is found to be proportional to N, where N is the atomic number. The values found agree with Hartree's calculations of the successive ionisation potentials of certain atoms.

1. The complete Denjoy-Fourier integral is of the form

where

the generating function ƒ(t) being integrable in the general Denjoy sense in every finite interval, and the integrals (1·2) being interpreted as principal values or in some other suitable way.

While the paper “On the pedal locus in non-euclidean hyperspace” was in the press, Professor H. F. Baker kindly directed the writer's attention to a reference from which it appeared that the euclidean case had first been studied by Beltrami. After publication, it was discovered that the main subject of Beltrami's paper was the non-euclidean case. He proves, by analytical methods, theorems which may be stated as follows: Let A0, A1,…, An denote the vertices of a simplex, [A], in non-euclidean space of n dimensions, and P a point such that the orthogonal projections of P on the walls of [A] lie in a flat, p; then the locus of P is an (n − 1)- fold, W, of order n + 1, which is anallagmatic for the isogonal transformation q. [A]; the isogonal conjugate of P is the absolute pole of p, and the envelope, w, of p is therefore the absolute reciprocal of W. Beltrami does not note the theorem, fundamental for the geometrical treatment of the subject, that W is the Jacobian of a certain group of n + 1 point-hyperspheres. He goes on to show that in the euclidean case the locus W breaks up into an n-ic (n − 1)-fold and the flat at infinity, while the envelope w does not, in general, break up. Finally, he notes that in two dimensions there is also a special non-euclidean case, in which W breaks up into an order-conic and a line, and w into a class-conic and a point; and that the appropriate condition is fundamentally that which is necessary for the degeneration of a certain class-conic into a pair of points. “But what are the points? And what is the corresponding condition satisfied by the absolute conic? This is a question which it would be interesting to resolve.” It does not appear that the subject has been pursued further; and in the present note an attempt is made to discuss fully the analogous case qi degeneracy in n dimensions.

This communication is a sequel to a former paper on “The General (m, n) Correspondence” read before this Society in March, 1926*. It contained the general exposition (§ 1) and the theory of complete and closed sets (§ 2). For convenience these reference numbers will be here adopted, so that the present work, which opens with § 3, will be understood as a direct application of the previous sections.