Part I The equations of a reacting gas mixture and their application to flames are briefly reviewed; ordinary diffusion is fully taken into account in terms of binary diffusion coefficients, but the diffusion thermo-effect and radiation are neglected. A one-dimensional flame model is considered whose equations are written concisely by means of suitable dimensionless quantities and temperature is chosen as independent variable. It is assumed that the flame velocity is not given, so that a related unknown constant, determined by the boundary conditions, occurs in non-linear differential equations. These are to be solved simultaneously with the complicated diffusion equations, and a general method of solution is aimed at. With a view to later generalization, the equations of the simplest ideal flame of reaction A $\rightleftharpoons $ B are discussed in detail. It is found that they are most conveniently solved in two stages: (i) by successive approximations to the solution of an integral equation, corresponding to the case of zero energy flux, and (ii) by a perturbation method, based on this 'unperturbed' case, which may be applied directly or, in general, in terms of parameter expansions. Part II Results of some physical and mathematical interest are given of computations relating to special ideal flames which separately bear some of the features of real ones. For example, in the case of an ideal flame in which there are two simultaneous reactions, the problem is solved by first assuming one of these reactions to be in equilibrium and then attacking the complete problem by considering the reaction rate previously neglected as a perturbation. The treatment of the general one-dimensional flame problem, proposed in part I, is then continued, and it is shown that the methods used there for the solution of the simple A $\rightleftharpoons $ B flame can be readily generalized as follows: (i) The complexity due to diffusion can be separated from that due to the eigenvalue character of the problem; the equations should first be solved with a set of certain well-defined 'ideal' binary diffusion coefficients-corresponding to a vanishing energy flux-and it will then generally be possible to arrive at the solution of the complete problem, with the actually given set of diffusion coefficients, by a perturbation method involving only linear equations, based on the 'ideal' or 'unperturbed' case. (ii) This 'ideal' problem can be made to depend on the solution of a single non-linear 'eigenvalue' differential equation (which is treated as an integral equation) by successive approximations, all other equations serving as mere definitions for better approximations to terms occurring in this fundamental equation.