﻿the Insolation of an Atmosphere. 885 



§ 7. These formulae offer several points of interest. As 

 a increases from to \ir and cos «->0, T tends to a definite 

 non-zero limit, although T l tends to zero ; T steadily de- 

 creases as a. increases, the limit being given by crT 4 = j7r/?S ; 

 Too tends to zero. It appears, then, that for sufficiently 

 oblique incidence the boundary is warmer than the interior. 

 Consider now the temperature distribution given by (28"). 

 When cosa = !?<, B(t) is constant and equal to \n& or 

 S cos a, and the state is isothermal everywhere ; and when 

 cosa<-^, the temperature steadily decreases inwards in 

 the interior. In spite of this there is at each point a 

 net flux in the outward direction ; so that here we have 

 a case where the net flux is in the opposite direction to the 

 temperature gradient. This would seem to be a novelty 

 in the theory of radiative equilibrium. (It is easy to 

 assure one's self that no contradiction with the second 

 law is involved.) These results are based oniy on the 

 approximate formulae (28") and (29"), but further investi- 

 gation confirms them. It is easy to see in a general way 

 how these curious temperature distributions arise. When 

 the solar radiation is nearly tangential, its effective intensity 

 is very weak, but owing to its obliquity it is entirely 

 absorbed in a thin layer close to the surface (provided 

 n is not zero). This layer is enabled to assume a definite 

 temperature, but no residual radiation penetrates to the 

 interior, which remains near the absolute zero. The out- 

 ward net flux is maintained at any point in virtue of the 

 outward radiatiou from the large amount of cold material 

 inside the point overpowering the inward radiation from 

 the small amount of warm material outride it. In the 

 limit when a = -j7r, the distribution of temperature is dis- 

 continuous ; the temperature is zero everywhere, except 

 at points in the surface. 



§ 8. Effect of rotation. — These results can only be applied 

 to a thick spherical atmosphere on the assumption that the 

 solar energy incident on any one place is all re-radiated 

 from that same place. Making this assumption, let us 

 tentatively take into account the effects of rotation. We 

 will calculate the time mean of the temperatures in any 

 given latitude X on the assumption that the axis of rotation 

 is perpendicular to the ecliptic. If </> is the hour-angle of 

 the sun, its zenith distance a. is given by cosa = cos<£ eos X. 

 Taking (29") as giving the "instantaneous" temperature 

 during the day and taking the latter as zero during 



