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An automated air shower array to investigate the properties of showers accompanied by high energy muons has been constructed to run in conjunction with the Durham magnetic automated muon spectro graph (M.A.R.S.). Measurements of the lateral distribution of muons have been obtained for muon energies ≥50, ≥100, ≥200, and ≥300 GeV in showers of size 5 x 10(^4) - 3 x 10(^6) particles for core distance of up to 50 metres. Details are given of the construction of the array, the computer handling of the collected data, its treatment and its analysis. Based on the current high energy nuclear interaction models, theoretical predictions of the muon lateral distributions have been examined and it is shown that the assumptions on which these models are based may be valid for high energy muons in E.A.S. Broad agreement is found with the predictions of the O.K.P. and the Slow Multiple Fire Ball (S.M.F.B.) models as applied to extensive air showers by Olejnizhack (1975) and Greider (1977). A comparison of the data is made with the predictions of the Feynman scaling model (Goned, 1975, and Fishbane et ,al, 1974) and it is shown that the scaling model and an assumed pure proton primary predicts lower values of muon densities than those observed in the present experiment. In contrast, it would appear that a combination of a scaling model and an assumed iron primary spectrum gives a better agreement with experimental data. In a preliminary experiment a small air shower array around the spectrograph was constructed and used to measure the sea level rate of muons having energies ≥5 GeV in small air showers. Experimental results are also presented on the sea level rate of these showers, their angular distribution, density and size spectra. In this experiment, comparison is made between the experimental data and theoretical predictions and the results., of other workers wherever possible. An account of the properties of the primary cosmic radiation and the showers they initiate is given in Chapters 1 and 2 with special reference to the muon component of E.A.S.