At present, the problem of thermoelasticity of shells with a heat-insulated crack, taking no account of heat transfer at the lateral surfaces of the shell, has been studied in sufficient detail [3]. Taking such heat transfer into account allows the stress--straln state of thln-walled shells in real conditions of operation to be more accurately investigated. In the present work, a method of solving the thermoelasticity problem for isotropic shells of arbitrary Gausslan curvature including a rectilinear heat-insulated cut is proposed. The shell is in thermal contact with an external medium, the temperature of which is a function of the coordinates. Consider a thin isotropic shell of thickness 2h. A triorthogonal coordinate system (xx, x2, xa) is applied, with axes directed respectively along the lines of principal curvature of the median surface and along the normal to that surface. The shell is weakened by a rectilinear cut L (Fig. i). The initial thermoelastic state is expressed as the sum of two states x j = m ,,x4 +N'(x,,xJ, where Hm(x,, x2) and H~ xa) are the components of the perturbed thermoelastic state due to the presence of the cut and the thermoelastic state in a continuous shell. Since the methods of solving heat-condUction and thermoelasticity problems for shells without a cut are well known [4], the question of determining H*(x,, x~) is considered here. To find the components of the perturbed thermoelastlc state, the equations of hollowshell theory are used [4]. With a linear temperature distribution over the shell thickness and convective heat transfer with the surrounding medium, according to Newton's law, the integral characteristics of the temperature (T,, T2) should satisfy the following equations