Fundamental. equations in the form most suitable for use in turhomachines derived from, basic principles of aero-thermodynamics are given first and a general discussion of thezr applications to the aero-thermodynamic design of turboma,ohines is made. In order to investigate the detailed variation of the flow of gas in tha channel formed by two neighboring blades, a relative stream surface in the middle of the channel is considered. After a discussion of commonlp used as well as possible design specifications, two forms of equations in terms of a stream function are given, applicable to most of the desibns discussed. The preceding results are used to investigate tha detailed flow in a single-stage axial-flow gas turhine, a single-stage and a seven-stage axial-flnw compressor and a mixed-flow inipelier. in the case of the gas tuxhins having cylindrical. hub and casing walls, designed fox a free-vortex type of tangential velocity distribution it is found: (1) the flow paths of teases deviate from cylindrical. surfaces beoause of the radial blade fort。and aonpressilJility of the gases,(2) the radial slow is sensitive to the location of the radial element of the blades,and (3) a large negative gradient of axial velocity in front of the rotor is observed. In the case of the single-stage axial compressor having cylindrical csasing and hub walls and the hub xadius, both designed far symmetrical veloaitp diagrams seven-stage axial compressor having a constant casing radius and an increasing at all radii, the analysis shows: (1) the radialiy increasing angular momentum of the air partials and the curved hub contour have the predominating iufluenve on the flow pattern; (2) the oompressibility of the air does not ohanoe the shapes of the stream Lines greatly but has a Large effect on the magnitude of the agiaL velocity; (3) the effect of the radial twist of the blade of this type of design on the flow pattern is small; (4) the air moving radially inward in passing through the inlet guide vanes, coutinues inward in the first rotor and moves radially outward in the following stator; (5) the radially inward and outward motion repeats in the succeeding stages of a multi-tage compressor; (6) the effect of this oscillatory radial flow is largest in the inlet stage a multi-tage compressor. In the case of the mixea-flow lmpeller designed for all radial blade elements; the trend in the variation of static pressure a long the caring obtain e 1 in the analysis, agrees well with the experimental data and the analpais also egpiains the reason of a local pressure drop along the oaring abservad in the experiments. In the last section, a vomparison is made between the radial variation of flow velocities in the gap between stator and rotor obtained in the preceding analyses of axial turbamaahines and those obtained in the simplified-radial-equilibrium calculation and in an approximate method based an sinusoidal radial-flow paths. The comparison shows that the first approximate methol gives considerable errors whereas tha second approximate method gives very accurate results.