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This paper is mainly a re-presentation of the magneto-ionic theory for oblique transmission through the ionosphere with special reference to numerical applications, but it is directed particularly to the problem of using ionospheric knowledge to plan short-wave long-distance communication. The quartic equation of the theory is solved graphically by the intersection of a straight line with a certain curve, and a chart is described for the construction of the latter for various values of the electronic density. By the method of stationary phase, simple expressions are obtained for the differential coefficients of the ray path, including the group time, lateral deviation and specific attenuation. Numerical integration along the ray path for a given density distribution is discussed with special reference to the relation of oblique to vertical transmission. Some preliminary results are given, but no detailed numerical survey is included, as this would more fittingly form the subject of a separate paper.

This paper is mainly a re-presentation of the magneto-ionic theory for oblique transmission through the ionosphere with special reference to numerical applications, but it is directed particularly to the problem of using ionospheric knowledge to plan short-wave long-distance communication. The quartic equation of the theory is solved graphically by the intersection of a straight line with a certain curve, and a chart is described for the construction of the latter for various values of the electronic density. By the method of stationary phase, simple expressions are obtained for the differential coefficients of the ray path, including the group time, lateral deviation and specific attenuation. Numerical integration along the ray path for a given density distribution is discussed with special reference to the relation of oblique to vertical transmission. Some preliminary results are given, but no detailed numerical survey is included, as this would more fittingly form the subject of a separate paper.

The most convenient research method of measuring completely the direction of arrival of waves reflected at the ionosphere appears to be that in which the phase differences of the signals received in an assembly of aerials are measured. If two pairs of similar aerials erected on lines perpendicular to each other are used, two independent phase angles may be obtained from which both angle of elevation and azimuth may be deduced.The apparatus described uses spaced coaxial loop aerials at a separation of 100 m. The signals from the aerials in a pair are amplified by means of matched receivers. The phase difference between the output signals from these receivers is displayed direct on a cathode-ray tube as the angle of inclination of the trace. With pulsed signals emitted from a suitable transmitter and with corresponding timing equipment in the receiver, the individual rays making up the total ionospheric signal may be separated from each other. The apparatus covers the frequency band 4–15 Mc/s, and the r.m.s. error of phase measurement is about 1°. Site errors, however, set a more severe limit to the accuracy of the directional measurements than do instrumental errors, and in practice it is found that, for example, over an oblique path corresponding to a range of 700 km, bearings can be measured with an accuracy of about 1° while the angle of elevation can be measured with an accuracy better than about 1½° so long as it exceeds 30°.These limitations mean that angles of elevation of E-layer reflections cannot be measured accurately at long range; it is possible, however, to obtain measurements of useful accuracy of the angle of elevation of F-layer reflections at ranges up to 1000 km or more. Bearings can be measured accurately at all ranges and for all reflections. The apparatus has so far been used principally for the study of F-layer reflections.

In the last few years a large part of the radio research at the Cavendish Laboratory has been concerned with the propagation of waves of low and very low frequency. The paper constitutes a summary of the results of the various experiments, which are described in detail in separate papers, some of which are as yet unpublished.^{3–10}The results of various independent methods of measuring the apparent height of reflection of the waves show that waves of 16–30 kc/s are reflected as if from a sharply bounded horizontal surface situated at a height of (72 ± 3) km when the sun is overhead. The apparent height of reflection varies regularly with the angle of the sun and its variation may be summarized by an equation. The waves of frequency 30–150 kc/s appear to be reflected from a height of about 75 km at oblique incidence, but there is some evidence that they may be reflected from as much as 10 km higher at vertical incidence.The polarization of the waves at all frequencies is found to be approximately circular at steep incidence, but at oblique incidence (65 ) waves of a frequency of 16 kc/s are linearly polarized. No measurements of polarization at oblique incidence have been made on the higher frequencies.The absorption of the waves changes very rapidly with frequency—on a summer day the conversion coefficient varies from about 0.15 at 16 kc/s to 0.002 at 70 kc/s. Important differences in behaviour near sunrise are observed on all frequencies at steep incidence and oblique incidence. The effects of a sudden ionospheric disturbance on the reflected waves are discussed and interpreted as implying a decrease in the apparent height of reflection; the amplitude of the reflected wave is scarcely altered on 16 kc/s, but is much decreased on higher frequencies.Finally, the present state of the theory of reflection of very long radio waves is discussed very briefly.