The problem of designing the best (optimum) liquid rocket engine (LRE) nozzles was formulated as early as 1931 by Glushko (see [1]).At the beginning of the fifties design offices began using parabolic or circular arc nozzle contours. The variational problem of selecting a nozzle with minimum dispersion losses was first solved in the fifties [2--5]. In [2] the solution of the variational problem was reduced to the numerical integration of a system of differential equations. In [3], independently of [2], a similar system of equations was obtained for a gas with a constant specific heat ratio and an effective means of analytic integration was found. In [4] the same integration was carried out for the generalized case of an arbitrary dependence of pressure on density. The principal result of [4] was repeated in [5]. Below, nozzles designed on the basis of [2--5] are called nozzles with a variational characteristic; they were developed in a number of design offices and used in working engines. In the American literature nozzles with a variational characteristic are called Rao nozzles after the name of the author of [5].At the Institute of Heat Processes (NIITP) in the fifties so-called tnmcated nozzle contours with a uniform characteristic were widely investigated. With respect to its thrust characteristics this one-parameter family of contours is similar to the nozzles obtained by variational methods, which is attributable to two circumstances. Firstly, for plane flows the solution of the variational problem is a truncated nozzle with a uniform characteristic. Secondly, for the axisymmetric case an untruncated nozzle with a uniform characteristic is the limiting solution of the variational problem.The investigation of the dispersion losses for both contour families has shown [6] that axisymmetric nozzles with a variational characteristic are approximately 0.1--0.2% better than truncated nozzles with a uniform characteristic. For short nozzles, as follows from calculations made at the t~nergomash NPO (1970), the difference may be much ~eater.It should be noted that although the dispersion losses, associated with the nonuniformity of the flow in the nozzle exit section, are one of the principal components of the impulse losses in the nozzle, the losses due to friction in the nozzle boundary layer are also important and usually exceed the dispersion losses.Nozzle optimization with allowance for the friction and dispersion losses in the variational formulation is an extremely complex and laborious task. One of the fast attempts was an investigation carried out by the t~nergomash NPO and the Computer Center of the USSR Academy of Sciences (1971), in which the general Lagrangian multiplier method proposed in [7] was employed. As a result of the calculations it was found that friction has only a very weak influence on the optimum contour.In the research carried out at the NI1TP [6] using direct calculation methods, without solving the variational problem, it was shown that the optimum contours having minimum friction and d...