Theory of Ionization by Cumulative Action of Successive Impacts by Electrons and by Quanta of Resonance Radiation.---The phenomena of the low-voltage arc seem to require, for their explanation, ionization of the gas molecules in two (or more) successive stages. In the theory here developed it is supposed that each electron from the cathode after falling through a minimum potential difference reaches an active zone in which it collides either with a neutral molecule and partially ionizes it or with a partially ionized molecule which thereby becomes completely ionized. Each molecule when partially ionized emits a quantum of resonance radiation and this may go off in any direction and will in a short distance be absorbed by a neutral molecule which thereby becomes partially ionized and emits another quantum of radiation. So the passage through the gas of the quanta of resonance radiation initially set free by electronic impacts is analogous to the diffusion of foreign gas molecules. Equations for the case of coaxial cylindrical electrodes are derived for the proportion of molecules partially ionized (1) by direct impact and (2) by resonance radiation in terms of quantities which can all be readily measured except $\ensuremath{\tau}$ the mean life of the ionized molecules and $\frac{1}{\ensuremath{\rho}}$ the scattering coefficient for the radiation, for which (except $\ensuremath{\rho}$ for mercury) only upper and lower limits can be estimated. The substitution of experimentally determined values leads to the conclusion that, under normal circumstances, partial ionization by photo-impact is many times as great as that by electronic impact alone and is necessary and sufficient to account for the observed ionization. The possibility of complete ionization by successive photo-impacts alone is also discussed.Theory of Low-voltage Arc.---If ${n}_{0}$ is the number of electrons emitted and ${P}_{0}$ is the proportion of partially ionized molecules in the active zone, ${n}_{0}{P}_{0}$ is the number of molecules ionized per second. Since the positive ions move more slowly than electrons, each positive ion neutralizes the space charge due to $4\ensuremath{\surd}(2\ifmmode\times\else\texttimes\fi{}1840M)$ electrons; hence the electronic current increases to $\frac{1}{(1\ensuremath{-}242{P}_{0}\ensuremath{\surd}M)}$ times the value it would have without ionization, provided the current is limited by the space charge around the cathode. As ${P}_{0}$ is increased by increasing the temperature of the filament or the applied potential, the current first increases to the saturation thermionic current, then the negative space charge is replaced by a positive space charge, the potential drop becomes concentrated near the cathode, the temperature of the cathode is raised by bombardment by positive ions, increasing ${n}_{0}$ and hence ${P}_{0}$; as a result the current increases at an accelerated rate, instability is usually soon reached and the arc strikes suddenly. The chief function of the gas is to give the positive space charge around the cathode which is the distinguishing feature of the arc. If $n$ is the saturation value of thermionic emission, the maximum current will be $\frac{3}{2ne}$ or $2ne$ depending on whether the voltage is below or above the minimum ionizing potential.Low-voltage Arc in Mercury Vapor.---Recent experiments indicate that the striking voltage is about 5.6 instead of 4.9 volts, and that the arc may be dependent on either the 4.9 volt (2536 \AA{}.) or the 6.7 volt (1849 \AA{}.) radiation according to the age of the vapor. Thus there seem to be two meta-stable states of the neutral mercury atom.Theory of Temperature of Ionization of Gases.---It is pointed out that resonance radiation must also be the chief factor in temperature ionization both in the electric furnace and in the sun and other stars.