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This thesis constitutes a study of the current limitation phenomenon of a low pressure arc discharge in mercury vapour. A new theory is presented which provides a theoretical relation between the D.C. current limit, IL and the independent discharge parameters of gas density p and tube radius r0. This theory is based on the fact that arc existence is impossible for values of the lumped parameter (pr) less than a critical value. If at I = 0, p = p0 , where p0 is such that (p0r0) is above the critical value, then by increasing the current to a sufficiently high level (i.e. to I = IL) such a degree of neutral rarefaction is brought about (due to the high level of ionization) that the local p is lowered to the point where (pr ) is equal to the critical value. Similarly a lower current limit is found, for as I is decreased, the wall sheath grows, effectively reducing r to such a point that (pr) reaches the critical value. Arc existence is only possible between these two limits. The upper and lower limit converge at a certain finite pressure, and below this, no arc existence is possible whatsoever. Proper experimental testing of such a theory, derived from fundamental relations, requires very careful control of the physical variables, especially local neutral density. Accordingly a special discharge tube, 5 metres in length, was constructed and the experimental results obtained therefrom were found to be in excellent agreement with the new theory. Theoretical predictions of the radial variation of neutral density for all values of I, were confirmed by experimental measurements of density using the absorption of resonance radiation. The case of constricted positive column geometry is considered, where a double sheath is formed at the diameter change. It is argued that if the double sheath is followed by a long uniform column, then the current limit IL, will be that of the smaller diameter tube, even though severe neutral depletion occurs near the double sheath. Experimental evidence is presented in confirmation of this prediction. Other aspects of the current limit theory are confirmed by Langmuir probe measurements and by heavy particle wall flux measurements using ionization gauges. These latter measurements also shed further light on the unresolved phenomenon of current-sustained pressure effects in the positive column. In the positive column at high currents, the neutral velocity distribution is found to be no longer Maxwellian (due to selective ionization of fast particles and of those with long flight paths) which leads to the unusual case of a non-Gaussian radiation absorption profile, although the only broadening influence is that due to the Doppler effect. The resultant profiles are computed and experimental results are presented which substantiate the calculations. The possibility of Hg sputtering by Hg ion bombardment of liquid Hg surfaces on the positive column walls is considered and a possible application to Hg valve technology is suggested. Measurements of metastable particle densities are presented and related calculations indicate the role of cumulative ionization in low pressure discharges. The Bohm sheath criterion for ions is generalized to include a number of complications, including that of oblique ion motion relative to the sheath edge. The plasma-sheath boundary is found to be mathematically equivalent to the Mach surface of fluid flow.