In 1849, Cayley and Salmon discovered that a general cubic surface in projective space of three dimensions over the complex numbers has twenty-seven lines on it. They remarked that all the properties of the twenty-seven lines would not become apparent until a better notation than they had given was found. This notation was discovered by Schläfli in 1858 in the double-six theorem (henceforth referred to as given five skew linesa1, …, a5with a single transversal b6such that no four of the ai lie in a regulus, the four ai excluding aj have a second transversal bj and the five lines b1, …, b5thus obtained have a transversal a6—the completing line of the double-six. The other fifteen lines of the cubic surface are then , where ai bj is the plane containing ai and bj.

To find the Centre of Gravity of a Ciroular Arc.

Let a (Fig. 12) be the radius, 2a the angle at the centre, AB the arc, of total mass m, G its centre of gravity symmetrically situated.

Imagine the arc to be part of a circle of string rotating uniformly with velocity u round C and of linear density ρ

Let Fn = 〈 a1,a2,…,an〉 denote the free group of rank n, and let θ denote the automorphism of Fn which permutes the generators cyclically, in other words:

This paper concerns an application of an algorithm for the second derived factor group as described by Howse and Johnson in [3]. This algorithm has as its basis theFox derivative (see [1]), a mapping from the free group F to the group-ring ℤF, definedas follows: let X be a set of generators of a group G, and let w = y1…yk with each yi∈X±1.

We study the following problem: establish existence and classification of closed curves which are critical points for the total curvature functional, defined on spaces of curves in a Riemannian manifold. This problem is completely solved in a real space form. Next, we give examples of critical points for this functional in a class of metrics with constant scalar curvature on the three sphere. Also, we obtain a rational one-parameter family of closed helices which are critical points for that functional in ℂℙ2 (4) when it is endowed with its usual Kaehlerian structure. Finally, we use the principle of symmetric criticality to get equivariant submanifolds, constructed on the above curves, which are critical points for the total mean curvature functional.

The generalised Stieltjes transform of a function f(t) ∈ L(0, ∞) is defined by

The integral converges for complex s in the region Ω, where Ω is the s-plane cut from the origin along the negative real axis.

1. My only justification for presenting this paper to the Society lies in the fact that, so far as I am aware, the uniform convergence of the Fourier Series is nowhere alluded to, and far less discussed, in any English textbook; while the precautions that are necessary in differentiating the series are hardly ever mentioned even in treatises which give a very thorough treatment of its convergence. I have confined myself almost exclusively to what may be called ordinary functions, as a complete discussion of what has been done in recent years for functions that lie outside the category of “ordinary” would make the paper much too long. For information as to the original authorities, I would refer to the paper which I communicated to the Society last session On the History of the Fourier Series. It is sufficient to say here that the proof I now give is simply an adaptation of that of Heine (Kugelfunctionen, Bd. I. 57–64, Bd. II. 346–353) and of that of Neumann (Über die nach Kreis … Functionen fortsch. Entwickelungen, 26–52).

Given a series Σan, we define , by the relation

where is the binomial coefficient . Let . If , the series Σan is said to be summable (C; k) to the sum s. If k > 0, p ≥ 1 and if, as n → ∞,

we say that the series Σan is summable [C; k, p] to the sum s, or that the series is strongly summable (C; k) with index p to the sum s. If denotes the difference , it is known that necessary and sufficient conditions for summability [C; k, p], k > 0, p ≥ 1, to the sum s, are that Σan be summable (C; k) to the sum s and that

In classical mechanics Gauss' Theorem for a gravitational field states that, if S is a closed surface and N the component of gravitational force along the outward normal, then

where β is the Newtonian constant of gravitation and M is the total mass inside S. This result has recently been extended to general relativity by E. T. Whittaker,1 who, however, considered only the case of a statical gravitational field, the line-element of which is given by2

where the coefficients U and αμν are independent of t. It is not immediately clear from his work whether the results are extensible to more general space-times.

§ 1. It is well known that, if , the convergence of sn to a limit implies the convergence of tn to the same limit. The converse theorem, that the convergence of tn implies the convergence of sn, is false. Mercer1 proved, however, that if , then both sn and tn tend to l. This theorem has recently been extended in various directions.2 In the present note the case of Abel limits is considered.

The study of non-associative algebras led to the investigation of identities connecting powers of elements of such algebras. Thus Etherington1 (1941, 1949, 1951) introduced the concept of the logarithmetic of an algebra, defining it roughly as “ the arithmetic of the indices of the general element”.

In “Viability Theory”, we select trajectories which are viable in the sense that they always satisfy a given constraint. Since the fundamental work of Nagumo [26], we know that in order to guarantee existence of viable trajectories, we need to satisfy certain tangential conditions. In the case of differential inclusions and using the modern terminology and notation of tangent cones, this condition takes the form F(t, x) ∩ TK#φ, where F(.,.) is the orientor field involved in the differential inclusion, K is the viability (constraint) set and TK(x) is the tangent cone to K at x. Results on the existence of viable solutions for differential inclusions can be found in Aubin–Cellina [2] and Papageorgiou [30,32].

The infinite products for sinx and cosx are most conveniently obtained in a rigorous way from the well-known factorial expressions for sinnθ and cosnθ which, when n is an even integer, take the forms

θ being put equal to and n made infinitely great.

Throughout this paper, we work in the category of (p-localized) spaces having the homotopy type of connected CW-complexes of finite type with base point. We consider a principal bundle

where Gn = SU(n), U(n) or Sp(n) and d = 1, 1 or 2 respectively. In this case, the bundle is obtained as an induced bundle by a mapping f of base space S2dn−1 from the classical group extension as follows:

The surfaces here considered were first discussed by Monge as surfaces whose normals are tangents to given developables. Under the name general surfaces moulures, Darboux treats them as the surfaces traced out by a fixed curve on a plane which rolls on a developable. When the developable is a cylinder he gives the general coordinates in his Leçons sur la Theorie Generale des Surfaces (Volume I., page 105). This led me to take up the more general case, but I later found that Darboux had also considered this in an ingenious and elegant manner in his Leçons sur les Systèmes Orthogonaux et les Coordonnées Curvilignes (Tome 1, pages 26–34). Perhaps this quite difFerent discussion will present some interesting points in analysis.

It has been shown in the previous three parts of this work, that the whole question of the convergence of the series solution, for the particular dynamical system under consideration, has turned upon the cubic equation

Before proceeding to generalize the results it will be shown how this cubic equation may be derived in a slightly different fashion.

Suppose λ is a positive number, and let , x∈Rd, denote the d-dimensional Gaussian. Basic theory of cardinal interpolation asserts the existence of a unique function , x∈Rd, satisfying the interpolatory conditions , k∈Zd, and decaying exponentially for large argument. In particular, the Gaussian cardinal-interpolation operator , given by , x∈Rd, , is a well-defīned linear map from ℓ2(Zd) into L2(Rd). It is shown here that its associated operator-norm is , implying, in particular, that is contractive. Some sidelights are also presented.