﻿the Insolation of an Atmosphere. 879 



thickness were zero. As regards the solar radiation, Gold 

 made an allowance for this by choosing t 2 so that the left- 

 hand side of (4) was slightly negative ; but again the 

 argument is unaffected. It appears then that Gold's 

 analysis, though doubtless giving the broad outlines of the 

 phenomenon, is inadequate in its details. 



§ 3. Now the complete phenomenon must be very com- 

 plex. Complications arise from the rotation of the earth, 

 the change of insolation with latitude, cloud-structure, 

 scattering, and the light from the sky, besides probably 

 the world-wide circulation of the air ; and the suddenness 

 of the upper inversion has always been to some extent 

 a difficulty. Instead of attempting to take account of 

 the various influencing causes simultaneously, it would 

 appear to be more in accordance with scientific method 

 to construct a number of idealized models, to work out 

 the theoretical solution for each separately, and then to 

 examine the extent to which the earth's atmosphere partakes 

 of their several characteristics. 



§ 4. The prohlem in principle. — As a contribution towards 

 this, it is proposed in this paper to consider the theory of the 

 radiative equilibrium of a mass of absorbing and radiating 

 material subject to insolation. The material is supposed to 

 be stratified in parallel planes, and to be subject at its outer 

 boundary to a parallel beam of incident radiation. The 

 latter will be supposed in the first instance to be normal to 

 the surface ; later we shall examine the effect of oblique 

 incidence. The material will be taken in the first instance 

 to be grey ; but later we shall suppose that there may be one 

 coefficient of absorption for the incident radiation, another 

 coefficient for the low-temperature radiation emitted by the 

 material itself. Further, we shall assume the material to be 

 infinitely thick, and to be in radiative equilibrium throughout 

 its mass. The assumption of infinite thickness involves little 

 or no loss of generality ; we could, if we liked, consider a 

 mass of finite thickness with an inner boundary consisting of 

 a black radiating surface, but since our results will only 

 involve the optical thickness, we need only suppose the ab- 

 sorption coefficient or the density to become suddenly very 

 large at an assigned depth in order to deduce the case of an 

 inner boundary from the solution for an infinitely thick slab 

 of material. 



The material being in a steady state must emit energy at 

 its outer boundary equal to the incident radiation. Across 



