Let us consider the thermoelastic component of stresses in the vicinity of a drillhole cut by flame.In order to determine the stresses acting near the face of the hole and responsible for the failure, we will use, as is usually assumed [i-3], the model of a homogeneous, isotropic half-space, around which a high-temperature gas stream flows, in which the temperature distribution has a domed shape. Here, the boundary conditions of thermal conductivity of the third type hold corresponding to the convective heat exchange which occurs during flame drilling [I].The use of the given model has led various authors to opposing results. Thus, in [I], analytical equations are presented, from which it follows that the normal stresses acting on areas perpendicular to the surface are compressed over the entire half-space. In [4], as in [I], a Gaussian axisy~netric temperature distribution in the gas stream is considered. However, the stresses indicated were obtained as compressive, not over the entire halfspace but only near the surface, and at depth they are transitional to tensile stresses.In studying the reasons for this contradiction, attention turns to the fact that the equations of [I] are approximate, and the error which they yield is not specified. By contrast, the conclusions of [4] are obtained by the numerical integration of integrals in the exact solution corresponding to the problem of thermoelasticity for the half-space. However, in [4], the method is not described for the elimination of features of a rather complex type in the expressions under the integral, which reduces the value of the operation, since features can, during numerical integration, introduce significant error, which has remained undetermined.Thus, it is difficult to express a priori a preference for one of these works [I, 4]. True, speaking in favor of [4], the author confirms, for example the results of [2,5]. However, in [2], heat exchange is not considered (the temperature of the half-space surface is fixed), and in addition, the error of the isolated asymptotes from which the stresses were calculated, was not estimated either. In [5], the planar problem was also solved by the finlte-element method, and although considerations o the plane problem are apparently sufficient to explain the basic qualitative patterns of thermal failure, the question of its applicability to the description of axisy~netrlc situations has not been established. Thus, the convective heating model for the half-space has not been studied to the end, and the search for simple equations for temperature and stresses in the half-space is of interest in both the planar and the axisyffi~etric case. These equations must be suited to the calculations and permit an esttmate of the errors in the latter, which play, as has been shown, a principal role. Moreover, the analytical expressions obtained may be used in solving more complex problems by the nu-~ricalmethod [3] for the purpose of verifying the correctness of the method and choosing its para-u~ters.Let the half-space z > 0 b...