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The present paper describes briefly a notation for representing continued fractions in many dimensions, which has the advantage providing a direct method of attack and of rendering intuitive, results which are usually proved by induction. The notation is the outcome of a generalisation which I previously made [1] in connection with the solution of certain difference equations. Only formal theorems are considered here. For a discussion of convergence reference may be made to the works [2, 3, 4, 5] cited at the end. The paper by Paley and XJrsell is particularly important since these authors discuss very fully the non-cyclic simple continued fraction

The application of the operational method of Oliver Heaviside to the solution of linear differential equations has been fully described in a recent Cambridge Tract by Dr H. Jeffreys.

The solution of linear differential equations by the method of Frobenius is a straightforward matter consisting merely in the substitution of a power series and obtaining the coefficients from a recurrence relation. The corresponding method when applied to difference equations is laborious and leads in general to divergent series from which convergent solutions can be obtained by a method due to Birkhoff(1). Actually a more appropriate method is to find a factorial series, but even here the direct substitution of such a series requires repeated transformations and throws little light on the structure of the equation. A method of symbolic operators was originated by Boole(2), but owing to the restricted definition of the operators it has a very limited scope. In the present paper a more general interpretation is given to the operators, and their application to a certain class of linear difference equations with rational coefficients is discussed.

The theory of the exact difference equation in the general linear case has been fully developed, but the corresponding theory for the non-linear equation of the first order does not appear to have been considered. In this paper necessary and sufficient conditions for the difference equation of the first order to be exact and the form of the primitive are obtained. It appears that two conditions are required for a difference equation to be exact, one of which is identically satisfied in the limiting case of the exact differential equation. These conditions are applied to determining the primitive in some cases where the conditions for exactness are not satisfied.

The operators used in this paper have the following definitions:— Δ ω u x = u x +ω - u x /ω, ∇ ω u x = u x +ω + u x /2. The summation operator has the property that, if u x = S x f ( t ) Δ ω t , then Δ ω u x = f ( x ), and lim ω → 0 S x f ( t ) Δ ω t = ∫ x f ( t ) dt . We also note that lim ω → 0 Δ ω u x = du x / dx , lim ω → 0 ∇ ω u x = u x . These notations are all due to N. E. Nörlund, their use allows us to consider general difference equations and to infer the corresponding results for differential equations.

SummaryThe homogeneous linear equation of the second order may be solved by means of a, continued fraction whose elements are coefficients in the equation. The two dimensional continued fraction is generalised to m dimensions and yields the solution of the equation of the mth order.

In a former paper the application of Boole's operators π and ρ to linear difference equations whose coefficients are rational functions was discussed. Reference should be made to this paper for the definitions of π and ρ and for Theorems I–V. In the present paper a more general type of equation is considered. Two new theorems are proved and used to obtain the formal solution and to establish the convergence.

The elliptic integral of the second kindcan, by substitution, be made to depend uponWriting E(K) = E, Jacobi's Zeta function is defined bywhereZ(u) is an odd function of u with period 2K.

A Set of numbers aij (called elements) arranged in a rectangular array of m rows and n columns constitutes a matrix of orders m × n. For example, of the arraysthe first is a matrix of two rows and three columns, i.e. of orders 2 × 3, the second is a square matrix of orders 3 × 3.From a square matrix of orders n × n we can form a determinant of order n. Thus from the above square matrix we can form the determinant of order 3

SummaryA general formula is obtained for the interference velocity when an aerofoil with elliptically distributed circulation is in a closed or open wind tunnel of any cross-section. The mapping of the section on the interior of a circle is given in terms of the Jacobian elliptic functions appropriate to the ellipse and rectangle. The result is worked out for an aerofoil which spans the focal distance in a tunnel whose section is an ellipse.

Consider a cylindrical or prismatic beam of isotropic material. Let the cross-section of the beam be a simply-connected region S bounded by the closed contour C.

Consider a cylindrical or prismatic rod of any form of cross-section bounded by plane ends perpendicular to the generators.

In this introduction we shall be concerned with a simple approach to tensor algebra, and the tensor formulation of stress, deformation and energy, leading to Hooke’s law, the properties of anisotropy, and considerations of elastic symmetry.

In this chapter we consider the action of forces applied to isotropic material which occupies the whole plane or a half-plane. In the latter case the forces can be applied by a body in contact with the half-plane, as in the case of buildings on elastic foundations and similar situations.

We now turn to the study of anisotropic material. It might be thought that the difficulties would be greatly enhanced but in fact, in the mathematical sense, the anisotropic is simpler than the isotropic case, the reason being that in the anisotropic case each term in the expression of the stress function is on the same footing whereas this is not so in the case of isotropy. Nevertheless the algebraic complexity of formulae is much greater. We might put the situation this way; in the anisotropic case formulae are obtained by simpler methods but are more complicated in algebraic structure.

Consider a tetrahedron, centroid P, of continuous material, isolated in thought from the material which surrounds it. Let the lengths of the edges be infinitesimal of order l.

This chapter aims to describe and unify the properties of the two functions of the complex variable in terms of which the solution of plane problems may be expressed in the case of isotropic material undergoing infinitesimal elastic deformation. As we are in effect dealing with plane geometry it is more convenient and intuitive to refer to the material as if it were in the form of a thin plate. In so doing there is no loss of generality, for we can proceed from this case of generalized plane stress to the case of plane deformation by a mere change in the value of the elastic constant x.

In this chapter the torsion problem for isotropic cylindrical or prismatic beams is considered. Methods of approaching the problem are examined and the general solution is obtained whenever the crosssection is simply connected and can be mapped on a circle. Applications of the principles of virtual displacements and virtual stresses to give approximate solutions are illustrated.

In this chapter we summarize the mathematical tools required to tackle the four fundamental problems stated at the end.

In this chapter we consider some aspects of linear anisotropy of general form and cylindrical anisotropy.