The compressible laminar boundary layer equations for a perfect gas in steady flow at arbitrary external Mach number and wall temperature distribution are solved approximately by the combined use of the Stewartson- Illingworth transformation and application of Lighthill's method to yield the shin friction and rate of heat transfer. Appendices are added which give the necessary modifications to the method for the separate cases of very low Prandtl number and for the flow near a separation point. A further appendix describes Spalding's method for improving the accuracy of the wall value of shear stress and rate of heat transfer distributions along a wall having a non-uniform temperature distribution.