The hyperbolic character of the differential equation satisfied by the velocity potential in linearised supersonic flow entails the presence of fractional infinities in the fundamental solutions of the equation. Difficulties arising from this fact can be overcome by the introduction of Hadmard’s ‘finite part of an infinite integral’. Together with the definition of certain counterparts of the familiar vector operators this leads to a natural development of the analogy between incompressible flow and linearised supersonic flow. In particular, formulae are derived for the field of flow due to an arbitrary distribution of supersonic sources and vortices. Applications to Aerofoil theory, including the calculation of the downwash in the wake of an aerofoil, are given in a separate report.