IntroductionNotationsPart I. Differential Equations of Compressible Flow and Properties of Their Particular Solutions2. Transformation of the Differential Equations3. The Particular Solutions of the Differential Equations4. The Properties of the Hypergeometric Functions of Large OrderPart II. Construction of the Solutions for Compressible Flow Around a Body6. The Functions for Incompressible Flow7. Conformal Mapping of Incompressible Flow on the Hodograph Plan8. Construction of a Solution about the Origin9. Analytic Continuation of the Solution Branch Point or Order 110. Continuation Logarithmic Singularity11. Transition to Physical PlanePart III. Improvement of the Convergence of Solution by the Asymptotic Properties of Hypergeometric Functions13. Asymptotic Solutions of the Hypergeometric Equations14. The Asymptotic Representation of F(av, bv; cv; τ) and F(av + β, bv + β; cv; τ)15. Transformation of the Function W(w; τ) Branch Point of Order 116. Continuation: Lo garithmic Singularity17. The Coordinate Functions x (q, θ) and y (q, θ)Part IV. Criteria for the Upper Critical Mach Number19. The Condition for the Limiting Line20. The Approximate Determination of the Upper Critical Mach NumberPart V. Application — Elliptic Cylinders22. The Functions z0(w), W0 (w) and Λ0 (w)23. Expansions of W0 (w) and Λ0 (w)24. The Stream Function ψ(q, θ)25. The Coordinate Functions x(q, θ) and y(q, θ)ConclusionsReferencesAppendix A Proof of Theorem (52)Appendix B Proof of Theorem (88)Appendix C Proof of Theorem (98)Tables of the Hypergeometric Functions