1895 - 96 .] Dr J. Halm on the Temperature of the Air. 261 
The coefficient of radiation is not the same for different elements ; 
it will certainly depend on their distance from the source of heat. 
Taking n different elements, we may express the corresponding 
coefficients by 
“A), a 2^0 J • » • • 
where a v a 2 . . . a n are numbers different from each other. The 
quantities of heat given to these elements by the soil are conse- 
quently 
- oqAo dm(f — t ± )dz ; - a 2 X Q dm(t' - t 2 )dz a n \dm(t' - t n )dz, 
and are obviously identical with the quantities of heat radiated 
to a v a 2 ... a n elements of mass with the same coefficient of 
radiation, A 0 . 
Applying this consideration to the whole of that part of the 
atmosphere which is influenced by the radiation of the soil, we 
conclude that we are allowed to divide the atmosphere into an 
unknown number of layers with unknown temperatures, but the 
same coefficient of radiation, A. The thickness of these layers natu- 
rally depends on our assumption of the element dm, and the 
nature of the function <f>( A). But, whatever be the differences 
between these, we can always assume an infinitely large number 
of sufficiently thin layers. 
Now, suppose there be 2 n layers, n being a large number, 
t v t 2 . . , t 2n their respective temperatures, we obtain the total 
amount of heat contributed by the soil to the whole atmosphere : 
- 2nX(f - T )dz, 
where T is the arithmetic mean of the temperatures of all the 
layers. 
Thus, we know that, whatever be the distribution of temperatures 
and densities throughout the atmosphere, the amount of heat given 
by the earth’s surface will only depend on the mean temperature, T, 
the temperature of the soil, t', and the coefficient of total radiation, 
2nX. It is evident that a supposed ideal atmosphere, with the 
same mean temperature, T, and the same coefficient of radiation as 
the real one, would produce exactly the same effect on the change 
of temperature in the earth’s surface. 
