A Fourier integral is said to be of finite type if its generating function vanishes for all sufficiently large values of ¦x¦. Because the coefficient functions are defined by integrals over a finite range, the behaviour of such a Fourier integral usually resembles closely that of the corresponding series.

In these Proceedings, Vol. xx (1920), pp. 198–204, the writer ventured (l.c. p. 203) on some remarks as to the reason why Sir G. Darwin and Mr Jeans had obtained discordant results for the stability of the pear-shaped figure of equilibrium, suggesting that this was due to a different mode of expansion of the functions involved. Sir G. Darwin used an infinite series of Lamé functions; Mr Jeans' method was equivalent to using the early parts from an expansion, of which every term, when expressed as an integral series of Lamé functions, would be an infinite series. At that time there was difficulty in obtaining Liapounoff's papers; since then, by the kindness of M. Belopolsky, of Pulkova, the whole of the four parts of Liapounoff's publication “Sur les figures d'équilibre peu différentes des ellipsoïdes d'une masse liquide homogène douèe d'un mouvement de rotation,” in all over 750 large folio pages, have become available; and these are now in the market. It is particularly interesting to see that the fourth part (1914) is devoted precisely to that change in the method of development which would arise in passing from Sir G. Darwin's expansion to the other expansion referred to above. And it is only by this change that Liapounoff is able to give the general proof of a form of expression of his results—in terms of polynomials and not infinite series—upon which his theorem of instability is made to depend. The careful consideration of the convergence of his expansions, which adds so greatly to the length of Liapounoff's papers, supplies materials for the proof that the expansion used in Mr Jeans' paper can be placed on a sure foundation, while Sir G. Darwin's expansion requires an estimation of the remainder.

§ 1. If the kinetic potential for the relative motion of two masses is written with an added constant as

a close connexion with the relativity quadratic appears. The latter is in fact

where

a modification of the primary form

which shows an unaltered determinant. The condition in respect to the determinant, suggested, I believe, by Schwarzschild, is one which to me appears to give the most significant form to the results. From the dynamical standpoint we may regard it as imposing a counterpoise in the inertia coefficients to the modification introduced by the potential; or from a geometrical point of view we may regard it as minimizing the departure from the normal use of coordinates. An illuminating example of the loss of meaning that accompanies transformation in which this condition is disregarded is furnished by the isotropic form which is sometimes given to Einstein's quadratic.

The following theorem, first enunciated by Dr Zeeman in 1899, has attracted the attention of several writers:

If five points A, B, C, D, E in three dimensions are such that the feet of the perpendiculars from E on the faces of the tetrahedron ABCD are coplanar the relation between the five points is symmetrical.

In the application of Elliptic Functions to the Theory of Numbers the two formulae of Jacobi

are of great importance.

The geodetic problem presented by a quadratic form is identical with the dynamical problem in which the same quadratic appears in the kinetic energy, and potential energy is not represented. If the quadratic in space-differentials is equated to kdt2, the problem is essentially the same whatever constant value is given to k; the corresponding feature is the arbitrary constant value of kinetic energy.

It is recognized that the main hope for advance in a rational theory of aeroplane propulsion lies in extensive use of the general principle of conservation of resultant momentum, even more than on considerations of energy; for energy can be dissipated by friction, while the relation of momentum to force is absolute.

The lines of a double-six will here be represented by the usual notation

where two lines whose symbols are in the same line or same column of this scheme are non-intersectors and all other pairs of lines intersect. Any six of the lines, no two of whose symbols are in the same column, and just three are in the same row, are generators of a quadric, and the actual position in space of each of the other six is determined by the two points in which it intersects this quadric.

The extension of Mellin's inversion formula expressed by the equations has been considered by Fowler who shows that some form of Stieltjes integral is essential to Poincaré's proof of the necessity of the quantum hypothesis. Fowler confines his discussion to a restricted type of function φ (y) which is sufficient for the physical problem. It will be proved here that the formulae hold with a general Stieltjes integral in the first equation.

The locus of a point the feet of the perpendiculars from which on the faces of a tetrahedron are coplanar is known to be a cubic surface having nodes at the corners: the pedal planes, by which I mean planes containing the feet of four perpendiculars, do not seem to have been much discussed. I propose to prove two theorems concerning them and make some remarks on their envelope.

§ 1. My primary aim here is to give some account of relativity in connexion with axial rotation, and in particular to deal with the disk problem. The justification of the use of a special quadratic form for this purpose suggests the propriety of some introductory remarks of a more general character.

This is a sequel to recent communications to the Society, on “The General (m, n) Correspondence” and “The (2, 1) Correspondence.” For convenience the reference numbers here adopted will be consecutive with what has preceded, so that the present work opens with § 6 as a direct consequence of previous results in earlier sections.

The motion of a light string loaded with equal masses at regular intervals has been discussed by operational methods. It is found that the system, though possessing only a. finite number of degrees of freedom, shows most of the characteristic features of dispersion. The operator e−σx/c that occurs in the discussion of uniform continuous strings is found to arise as the limit of an operator defined wholly in terms of definite integration, and its interpretation involves a theorem analogous to Taylor's theorem, but apparently more general. There is no phenomenon analogous to the conduction of heat. Any irregularity in the distribution of mass, however, produces strong internal reflexion of the shortest waves, and may provide a mechanism for conduction.

Engel and Stäckel in their Theorie der Parallellinien von Euklid bis auf Gauss (1895) call attention to the fact that Saccheri and Lambert have used in their argument the property of the triangle that the external angle is greater than either of the interior and opposite angles (Euc. I. 16), which is not true for triangles of all magnitudes when the hypothesis of the obtuse angle holds good. They do not show how starting from the point of view of Saccheri and Lambert the proofs should be amended so that they would be valid in the Elliptic Plane. Bonola (Non-Euclidean Geometry, translated by Carslaw, 1912, p. 31) in his account of the subject also uses Euc. I. 16. It is the object of this paper to obtain the main results proved by Saccheri and Lambert when the hypothesis of the obtuse angle holds without using Euc. I. 16. It is not suggested that this is a convenient method of procedure but it may perhaps have some historical interest. The key to the procedure here adopted is found in the reversal of the Euclidean order in a certain group of propositions, viz. I. 16, 18, 19, 20.

When a triangle ABC is in perspective with A′B′C′ and with B′C′A′, so is it also in perspective with C′A′B′, and the two triangles may then be said to be in cyclic perspective. The centres of perspective then form a third triangle XYZ, such that, of the three triangles, every two are in cyclic perspective with the vertices of the third for centres. The figure thus arising is the figure of Pappus' Theorem with nine points and nine lines incident by threes.

The present work is the outcome of a study of the special type of (3, 2) correspondence on a circle, namely that between a point and the extremities of its pedal line. This led to considering correspondences in general, and to the formulation of concepts which are believed to be new, for example the Canonical Forms, § 1 (2), and Multipliers, § 1 (3). The case of the (n, 1) correspondence has already been given by the author.

The circles in a plane can be represented by the points of space of three dimensions in such a way that the points (point circles) of the plane correspond to the points of a quadric Q in space and two orthogonal circles in the plane correspond to two points conjugate with respect to Q. An ∞2 system σ of circles in the plane corresponds to a surface S in space and the reciprocal Sσ′ of circles in the plane closely and reciprocally related to the system σ. So far as I know only trivial examples of such systems σ and σ′ have hitherto been noticed and my object is to direct attention to an illustration drawn from the modern geometry of the triangle which is less simple and perhaps more interesting.

The following is an alternative to Proposition II of the former paper. It does not assume the second of the initial assumptions in Art. 2, nor does it assume that the angles supplementary to equal angles are equal, which Hilbert regards as a proposition (Foundations of Geometry, p. 18 of the English translation).

§ 1. “Stretches” joining Points in a Metrical Space.

If x, y, z are any points in a metrical space, then by definition

Regarding x and y as fixed, we may define the set of points z for which

as the “stretch joining the points xy” or more shortly the stretch xy*. Then for points z not on the stretch,

The stretch xy always contains the points x and y themselves, which we may call the “end-points” of the stretch. In particular cases, it may contain no other points: e.g. with the function δ0(x, y) = 1 when xǂy, each stretch contains only its end-points †.

1. Soit donnée l'équation de Mathieu

a et q étant des paramètres réels.