When the voltage across a gap containing a gas is increased by only an extremely small, fractional amount in the "breakdown voltage" region, the gap changes from a very good insulator to a good conductor. Although the breakdown voltage may be accurately predicted by the Townsend mechanism, the actual breakdown process has been subject to continuing controversy. The purpose of this paper is to present results of a program at these Laboratories to make calculations of the breakdown transition with an electronic computer, the IBM 704.