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.

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.

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).

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.

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.

The object of these notes is twofold. First, to connect the theory of the possible shapes of Jacobi's ellipsoidal equilibrium figure, for a given mass of homogeneous rotating liquid, with the theory of its stability and other properties given by Prof. H. F. Baker in his paper “On the stability of rotating liquid ellipsoids” in Proc. Camb. Phil. Soc. vol. 20, pp. 190–97. And secondly, to indicate two inconclusive reasonings on the subject in Thomson and Tait's Natural Philosophy (2nd edition), arts. 778 and 778′, and to complete the proofs of the results in question.

The equation to be considered is of the type

where p (x) is continuous for all real values of x, even, and periodic. It is no restriction to suppose that the period is π, and this assumption will be made, so that the equation is virtually Hill's equation.

The fact that the Mathieu equation, and, as was proved in the preceding paper, the more general Hill equation, can admit of but one solution of period π or 2π, suggests the problem of determining how far the second solution deviates from periodicity when the first solution has either the period π or the period 2π

The constructs in question are represented by the vanishing of all determinants of two rows and columns drawn from a matrix of r + 1 rows and s + 1 columns, where r can be taken less than or equal to s. They are thus the multiple constructs of highest dimension on the well-known family of constructs generated by projective systems of linear spaces. In the present note the constructs are considered from a somewhat different point of view, the principal applications being to the determination of directrix loci of minimum order for the systems of spaces which they contain, and to the re-proof and extension of a known theorem.

Prof. W. P. Milne has recently been investigating, in connection with the general cubic surface, the Equianharmonic Envelope S4 and the Harmonic Envelope T6, denned as the envelopes of planes cutting the surface in equianharmonic and harmonic cubic curves respectively: he suggested to me that I should consider the particular case of the quadrinodal cubic surface.

The usual proof of the adiabatic invariance of the quantum integrals of a multiply periodic dynamical system rests on the assumption that the Hamiltonian equations remain valid for the adiabatically changing system, with the same Hamiltonian function H, or at any rate with a new Hamiltonian function differing from the old one only by terms that vanish with ȧ, (a) being the varying parameter.

The experiments described in this paper were carried out in the University of Sydney during November 1924 and, owing to the short time which was available, are of a purely preliminary character. The following account is given as they are not likely to be resumed in the immediate future.

In a paper by Slater a new theory of radiation is discussed and the following explanation of the breadths of spectral lines is given: “An atom in the ith. state has a probability Pi of suffering in unit time a transition. Thus there is a probability Pi that the vibrations of each of the oscillators will simultaneously cease. But we shall assume that, in addition to this probability Pi of ceasing its oscillation altogether, each oscillator has also an independent probability Pj of suffering an interruption in which it ceases its oscillation as if it were leaving the state, but immediately begins again as if it were entering the same state” (i.e. with an arbitrary phase difference). “This term Pj is the same as the probability that an atom in the jth state will leave that state.” The total probability of interruption of vibration is thus (Pi + Pj) which is symmetrical with regard to the two end states. This makes the breadths of absorption and emission lines equal and so satisfies Kirchhoff's law.

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.

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.