Many fundamental problems in solar physics are likely to remain unsolved until a spectroscopic resolving power of the order of 10 6 becomes available. Grating or prism spectroscopes have not yet been constructed to satisfy this need. The writer investigates the possibility of obtaining the required resolving power by interferometric means. The chief difficulty, in obtaining a high-resolution interferometric analysis of a continuous spectrum broken by absorption lines, lies in the fact that high primary monochromatization is required in order to present the interferometer with a range of spectrum small enough to be analysed without overlap of orders. This consideration sets a limit to the factor of gain obtainable by interference methods, and defines somewhat narrowly the correct conditions of employment of the spectroscope and interferometer. An experimental arrangement is described which is designed to satisfy these conditions. A short etalon of high reflectivity is placed behind the spectrum of the Sun formed by the Oxford Solar Spectroscope. Recollimated light from this spectrum is analysed by the etalon, and produces an interference pattern in a plane conjugate to that of the spectrum, in the focal plane of a camera. This pattern consists of a system of circular heterochromatic channels, and within these channels a system of monochromatic absorption fringes appears. These absorption fringes are interferometric spectra of the solar absorption lines, and their definition is determined by the resolving power of the etalon only. The sharpness of the fringes makes possible wave-length measurements of high precision. Measurements in the D region show that the probable error for a single plate will be about 0.001 A. The interference photographs display continuously, without overlap of orders, an interferometric spectrum exceeding many times the range of the etalon used. This feature, combined with the resolution, makes the method suitable for the photometric determination of line contours. Results of applications in this field will be presented in a subsequent paper.