The frequency response of the h.f. quadrupole probe is calculated to be used as a diagnostic tool for measurements of electron density and temperature. In §2 the magnetic field is assumed to be zero, and ion motions are neglected. For a Maxwellian plasma, the so-called ‘Landau wave approximation’ is compared with various more sophisticated treatments, such as numerical integration or super-Cauchy and multiple water-bag models. The range of validity of this approximation is shown to be large, and the results can be applied to the most interesting parts of the experimental observations. All results previously established are recovered with greater speed. Having studied various disturbances (collisions, inhomogeneity and relative motion of the probe with respect to the plasma), it is deduced that the best way to determine the electron temperature is to use the anti-resonances due to beating between the Landau wave and the cold plasma field. In § 3 we describe the quadrupole probe, launched in December 1971 as part of the CISASPE rocket experiment. To deduce the electron density and temperature from these measurements, it is necessary to consider the influence of a static magnetic field, such as the earth's magnetic field. The general case could be treated by numerical integration, though with great difficulty, but it is shown that in most ionospheric conditions, in the vicinity of the upper hybrid frequency ωT the above treatment is again possible, the plasma frequency simply being replaced by ωT, and the thermal velocity slightly modified. These assumptions are used to deduce the electron density and temperature profiles.