The present paper is concerned with formulae by which double integrals of functions of two independent variables may be evaluated approximately. The number of such formulae published hitherto is not great, and it has seemed desirable both to make a systematic search for new formulae, and to test the comparative merits of these, and of those previously known, by computing the numerical values of certain selected integrals.

The equation of the propagation of electric signals along cables, generally known as the equation of telegraphy, may be written

Particular solutions of this equation, adapted to various purposes have been found by Heaviside, Poincaré, A. G. Webster, T. W. Chaundy, § and others. The object of the present paper is to unify the theory of the equation by exhibiting the relations which these solutions bear to each other, and by obtaining them as particular cases of a general solution. The derivation of new particular solutions by the solution of integral equations is also discussed.

In the Proceedings of the London Mathematical Society, Ser. 2, Vol. 20 (1921), pp. 465–489, Professor H. W. Turnbull has studied the projective invariant theory of three quadrics. The following paper is based on this work and develops one definite section of the theory. From the geometrical point of view the linear complex is now seen to be fundamental in the study of three arbitrary quadrics; particularly when their (2, 2, 2) invariant φ123 vanishes.

be the binary cubic whose coefficients a1, are complex numbers represented on the Argand Plane. Then if its roots are z1, z2, z3, the three corresponding points form the vertices of a triangle A1 A2 A3. Let this triad of points be said to represent the cubic. Then its Hessian

is represented by a certain pair of other points; likewise every first polar

associates a definite pair of points (z) with any given point (y).

The numerical solution of Integral equations with variable upper limits has been investigated by Professor Whittaker. In this investigation the nucleus, supposed to be given numerically by a table of single entry, is replaced by an approximate expression consisting of a finite number of terms, each term involving an exponential or simply a power of the variable, and then the solution is found as an analytical expression from which its numerical values may be computed. The numerical solution of integral equations with fixed limits has been discussed by H. Bateman.

The Partial Differential Equations of Physics may be defined as those equations which can be derived from a “least action principle,” that is, as those which are obtained by making a certain integral stationary by the methods of the Calculus of Variations. But, generally speaking, such equations belong to conservative physical systems, and not to those which involve dissipation of energy. In this note it is shewn that a certain class of dissipative equation, of which the best known example is the equation of telegraphy, can be derived from such a calculus of variations problem.

The first part of the following investigation was begun before the discovery that Mr E. Kasner had already touched upon the apolarity theory of double binary forms in an important work on the Inversion Group (Transactions of the American Mathematical Society, Vol. I (1900), pp. 471–473). The theory is carried further in what follows, with special reference to the (2, 2) form. The second part answers questions raised by Professor A. R. Forsyth in the Quarterly Journal, 1910, p. 113. It appears that the general (2, 2) form admits of three independent automorphic transformations, but the general (n, n) form admits of none, if n exceeds two.

If

represent the points A, B and C, D respectively and if

then A, B, C, D are four harmonic points on a circle. If 0 is the mid point of AB then OA2 = OC. OD and ∠COB = ∠DOB.

In a previous paper the author has employed the expansion

where

and

to establish the theorem that, if −π/2<amp γ<π/2, the function Ps/Ts+1 remains finite as γ→∞. This theorem is valid provided that z is not real and ≧ 1.

A Generalisation of Christoffel's Summation Formula. The Recurrence Formula

is valid for all values of n and m; but, if m is a positive integer, , hence, when n = m, (1) becomes

A Theorem of Dougall's. In Vol. XVIII. (p. 78) of these Proceedings Dr Dougall has established the theorem that, if m is a positive integer,

An elegant symbolic method of solving differential equations was developed by Heaviside in his “Electrical Papers” and “Electromagnetic Theory,” chiefly in connexion with problems concerning electric currents in net-works of wires. Attention has recently been called to the method by Bromwich, who applied it to a wider range of problems and gave an extension of Heaviside's formula; another generalisation of the formula has been obtained by Carson.

In the present paper a formula is obtained which contains the formulae of Heaviside, Bromwich and Carson as particular cases, and whose form is such that it may be readily applied to physical problems.