The radius of convergence of the power series is 2π.

The function is regular within a circle whose centre is the origin of the z plane and radius 2π, and can be expanded in a Taylor's series converging at all points within the circle.

We shall use the notation Σnr= lr + 2r+…+ nr. Mr A. J. Gray suggested to me that n(n+ 1) is always a factor of Σnr, and that, in addition, 2n + 1 is a factor when r is even.

The greater part of this paper consists of generalisations of well-known theorems regarding perpendiculars to the sides of a triangle, or other base-lines, the perpendiculars being replaced by isoclinals.

The equations

where f1(t), f2(t)φ(t) are polynomials in t, determine a unicursal curve. If α and β can be found so that

then the point (α, β) is on the curve and is given by two values of t, t1 and t2, which are the roots of the equation at2 + 2bt + c =0.

Mathematical induction in its simplest form may be stated thus: Suppose there is a set of propositions p0, p1, p2…which are so related that the truth of pn implies the truth of pn+1 then if p0 is true, it follows that all the other propositions of the set are true. For since p0 is true therefore p2 is true, therefore p2 is true, and so on as far as we please.

The determinant

each row of wliich contains the same n elements in the same cyclic order, with ′a1 always in the leading diagonal, is the product of n linear factors, which we shall write as follows

where ρ is any primitive nth root of unity.

In the Theory of Potential the term Green's Function, used in a slightly different sense by Maxwell, now denotes a function associated with a closed surface S, with the following properties:—

(i) In the interior of S, it satisfies ∇2V = 0.

(ii) At the boundary of S, it vanishes.

(iii) In the interior of S, it is finite and continuous, as also its first and second derivatives, except at the point (x1, y1,z1).

In investigating the properties of the cubic curve C, we are led to consider its Hessian C′, its Cayleyan Γ and the Hessian of its Cayleyan Γ′. In the present paper we propose to deal with the intersections of the systems C + λC and Γ + μΓ′.

The following problem appears in Robert Simson's “Opera Quaedam Reliqua,” pp. 472–504 :

“ Si a duobus punctis datis A, B ad circulum positione datum CDE inflectantur utcumque duae rectae AC, BC circumferentiae rursus in D, E occurrentes; juncta DE vel continebit datum angulum cum recta ad datum punctum vergente ; vel parallela erit rectae positionae datae; vel verget ad datum punctum:” i.e. if from two given points A and B any two straight lines AC, BC are drawn to a circle CDE given in position, and they meet the circumference again in D and E, then the straight line DE (I.) will inake a constant angle with a straight line passing through a fixed point, or (II.) will be parallel to a straight line given in position, or (III.) will pass through a given point. This final form of the result was only arrived at by Simson after he obtained the aid of Matthew Stewart.

Internal Reflections in a Sphere. A ray of light PQ (Fig. 1) incident at any point Q on the surface of a transparent sphere is partly reflected and partly refracted along QR1. At R1 it is partly reflected along R1R2, and partly emerges along R1s1. The same thing occurs at R2, R3, etc. on which the successive reflected portions fall. AB is the diameter parallel to the incident light, and OQ the radius to the point of incidence. Let μ denote the index of refraction of the material of the sphere relative to the surrounding medium. Divide OA in C so that OC : OA = 1 :μ

General Construction for Refracted Ray. The two-circle method of finding the direction of a ray refracted at a plane surface is very old, but seems to be now almost forgotten. It is particularly convenient when the refraction of several rays is to be determined, and its application to the case of a prism is especially elegant and leads to a simple self-contained proof of the condition for minimum deviation.

If H (Fig. 1) be the middle point of a straight bar QP and if a straight bar OH of length one-half of QP be pin-jointed to QP at H, a simple linkage is formed, which may be called a T -linkage.