The object of the present paper is to establish the equivalence of the well-known theorem of the double-six of lines in projective space of three dimensions and a certain theorem in Euclidean plane geometry. The latter theorem is of considerable interest in itself for two reasons. In the first place, it is a natural extension of Euler's classical theorem connecting the radii of the circumscribed and the inscribed (or the escribed) circles of a triangle with the distance between their centres. Secondly, it gives in a geometrical form the invariant relation between the circle circumscribed to a triangle and a conic inscribed in the triangle. For a statement of the theorem, see § 13 (4).

The formula

where αp+1 = 1/2(m + n), αp+2 = 1/2(m-n), R(m±n)>0 and x is real and positive, was given by MacRobert (Phil. Mag., Ser. 7, XXXI, p. 258). From it the formula (6) below will be deduced.

§1. Introductory. The formula

where w is zero or a positive integer and | ζ | > 1, was given by F. E. Neumann “Crelle's Journal, XXXVII (1848), p. 24”. In § 2 of this paper some related formulae are given; the extension to the case when n is not integral is dealt with in § 3; while in § 4 the corresponding formulae for the Associated Legendre Functions when the sum of the degree and the order is a positive integer are established.

If the right-hand side of the expansion

is integrated m times from 1 to µ, it becomes

Given the series ,the n-th Casáro sum of order k is defined by the relation

where is the binomial coefficient . Let Then Σan is said to be summable (C; K) to the sum s if, as n → ∞, The series is said to be absolutely summable (C; k), or summable | is convergent. The series is said to be strongly summable (C; k) with index p, or summable [Ck, p], to the sum s if

It is assumed that k and p are positive.

The problem of solving the equation of thermal conduction for cases in which heat is generated in the interior of the medium under consideration arises frequently in physics and engineering. It occurs, for instance, when we consider the diffusion of heat in a solid undergoing radioactive decay (1) or which is absorbing radiation (2). Complications of a similar nature arise when there is a generation or absorption of heat in the solid as a result of a chemical change-for example, the hydration of cement (3). The particular case in which the rate of generation of heat is independent of the temperature arises in the theory of the ripening of apples and has been discussed by Awberry (4).

In this note we derive some integrals involving confluent hypergeometric functions and analogous to Lommel's integrals for Bessel functions. Although the method of derivation is straightforward, the integrals do not seem to be mentioned in the literature.

By applying Gauss's Theorem it can be seen that, if n is a positive integer and α is not integral,

so that

In section 2 it will be proved that, if

Where α and β are not integers

Where μ and M are constants independent of n.

The formulae

where γ = ½ (α + β + l), and

where γ = 1 + β, were given by Gauss (Ges. Werke, iii, pp. 225, 226). It is here proposed to find the corresponding expressions for the hypergeometric function when γ has general values (not zero or negative integral). These will be derived in section 2 by applying Lagrange's expansion

where

and that root of equation (4) in x is taken which is equal to λ when w = 0. Two generalisations of Whipple's Transformation will be obtained in section 3.

We consider a volume of material, divided into two regions 1 and 2. each of density ρ, by a moving surface S. On S a change of phase occurs, at a definite temperature (which we may take to be zero) and with absorption or liberation of a latent heat L per unit mass. If θl, kl, K1 are the temperature, thermal conductivity and diffusivity of phase 1, and θ2, k2, K2 corresponding quantities for phase 2, the surface S is the isothermal

and the boundary condition on this surface is

Subscript letters denote partial differentiation.

Let f(x, y, z, t) satisfy the equation

For certain purposes, particularly in connection with the propagation of a boundary of fusion etc., it is of interest to discover solutions of (1) which permit the equation:

to be solved explicitly in the form:

This suggests the examination of solutions of the type

where

and f, ϕ, ψ are functions to be determined. To save repetition, Roman capitals denote arbitrary constants throughout.