Pressure broadening causes lines in infrared absorption bands to have considerably greater halfwidths in the lower layers of a planetary atmosphere than in the upper layers. As a result, radiation emitted upward from the wings of lines in the lower atmosphere is not strongly absorbed by the upper layers. Such radiation is thus free to escape to the cosmic cold. In this paper we calculate the net loss of heat by radiation from the various layers in the stratosphere, which is greater for the lower layers than for the upper layers. This affords a new basis for the explanation of the existence of the stratosphere. The radiation budget for the stratosphere as a whole is reconsidered after these ideas and after new data on the transniission of the upper air first published herein. It is shown that the heat losses from the stratosphere by radiation from the 9.6 Oi, 15 j# CO2, and 50 H bands are approximately balanced by the heat gained from absorption of terrestrial radiation by these bands and by the absorption of solar radiation by Oi in the ultraviolet.