This paper represents the application of the Principle of Reciprocity, formulated in a previous communication, to the outstanding problems of classical and quantum electrodynamics.

The first step consists in the formulation of a reciprocally invariant Lagrangian function for a system of electrons in interaction with the electromagnetic field. A study is made of the unaccelerated motion of an electron, and this is subsequently extended to embrace the problem of an electron in arbitrary motion. It is found that the usual difficulties of classical electrodynamics do not appear. The methods of the earlier paper are applied to the derivation of the Hamiltonian energy of electron and field, and this enables a quantized formulation of the theory to be given, which also does not lead to the usual divergence difficulties.

The relaxation technique of R. V. Southwell is shown to be applicable in certain cases to transonic problems. For a uniform stream with a low subsonic velocity impinging on a symmetrical 2-dimensional double wedge, an asymmetrical supersonic region can be isolated in the neighbourhood of the corner of the wedge, and the streamlines and the values of the Mach number within this supersonic region can be determined with the aid of relaxation methods. Difficulties must be expected to occur in the neighbourhood of the sonic line, but in the present problem these have been surmounted.

A square matrix A = (aij) is expressed symbolically in terms of Clebsch-Aronhold equivalent symbols aij = aiaj = βibj = …, and the symbolic expressions for symmetric functions of the latent roots of A are considered, the relation between these functions and projective invariants of the bilinear form uAx being noted. The Newton and Brioschi relations between the symmetric functions are obtained by reduction of symbolic determinants and permanents respectively, and the Wronskian relations are shown to be equivalent to certain identities between determinants and permanents due to Muir. Also the fundamental theorem of symmetric functions is obtained symbolically as a consequence of the first fundamental theorem of invariants. The paper concludes with a note on the symbolization of the h-bialternants, that is of the traces of irreducible invariant matrices of A.

The method of iteration of penultimate remainders, introduced by S. N. Lin for approximating by stages to the exact factors of a polynomial, is subjected to theoretical analysis. The matrix governing the iterative process is obtained, and its latent roots and latent vectors are found. Incidental theorems yielding further factorizations are proved, and processes are developed for accelerating convergence. Numerical examples illustrate varying situations likely to arise in practice.