The downward longwave radiation flux at any level in the atmosphere can be calculated from ascent data, applying the theory of the transfer of the longwave radiation in the atmospher. However, such information is generally not available at sites where longwave radiation is needed for particular applications. For the surface downward longwave radiation, various experimental formulae have already been presented, all of which, though they agree only statistically with the observed fluxes, provide a valuable guide for the estimate of the downward longwave radiation under the given meteorological conditions. However, the choice of the type of formula is arbitrary and the coefficients of the formula depends strongly on data analysed. Brutsaert (1975) presented a new type of formula for the apparent emissivity of a cloudless sky in the unique way. It should be stressed that his formula is derived analytically on the theoretical basis by approximating the ascent profile by the exponential function. The purpose of the present study is to attempt an extension of Brutsaert's procedure and to derive a general formula which can describe the downward longwave radiation at any level in the lower troposphere both for the clear and overcast conditions.<br> The general solution for the downward longwave radiation flux at the level zr in the subcloud layer is given by _??_ where u_z is the reduced water vapor content, which is defined by _??_ eq. 2<br> where T, p and ρ_w are the air temperature, the atmospheric pressure and the density of water vapor respectively. Subscriptions 0, r and c denote the ground, the reference level and the base of cloud respectively. ε is the slab emissivity of the isothermal layer of the mixed gases of water vapor and carbon dioxide, which can be approximated well by ε=Au^m. eq. (3)<br> Ascent profiles of the atmosphere can be approximated well in the troposphere, especially in the lowest five kilometers, by following exponential functions<br> _??_, eq. (4)<br> _??_ eq. (5) _??_<br> where σ, g and R are Stefan-Boltzmann constant, the accelation of gravity and the gas constant of the dry air respectively. According to data provided by Robinson 1947 the coefficients A and m of equation (3) are 0.723 and 0.090 respectively. is the temper ature lapse rate, which is about 6.5 K/km in the standard atmosphere. The reason of choosing the exponential functions for equations (4) through (6) is that they vanish at the top of the atmosphere. Substituting equations (2) through (6) into equation (1) leads to the following analytical expression of the downward longwave radiation flux at any level in the subcloud layer as the function only of surface weather elements;<br> _??_ where B (a, b) and I_x (a, b) are the beta function and the incomplete beta function ratio respectively, the product of them being the incomplete beta function. When z is expressed in km, k₁ and k₂ are 0.695 and 0.605 respectively, where _??_, and _??_<br> Let us consider the special case that the reference level lies on the ground: substitutin g zero into zr in equation (7) leads to a formula for estimating the apparent emissivitY of an overcast sky as follows<br> eq. (8)<br> According to observations of the cloud height at Fukuoka (Tomitaka, 1957) 9 the mean base heights of low and middle clouds are 1.21 km and 4.13 km respectivelY. For an overcast sky of low cloud, equation (8) is rewritten in the limit when ze approaches 1.21km as follows _??_ eq. (9) and for middle cloud in the same manner<br> _??_ eq.