260 Proceedings of Royal Society of Edinburgh. [sess. 
of the soil, which alters its thermal capacity, as well as the quan- 
tity of heat lost by the change of water into vapour ; — all these 
phenomena, which must he taken into account, add to the diffi- 
culty of the investigation. The following sketch, however, may 
perhaps suffice to show how, by comparatively simple considera- 
tions, the effect of all the disturbances just mentioned may find 
expression in the mathematical solution of the problem. 
The first principal question we have to answer will be : By what 
means does the lowest layer of the atmosphere, the temperature of 
which is recorded by our thermometrical instruments, receive or 
lose heat 1 ? There can be no doubt that the chief cause of every 
change of temperature within this layer must be found in the 
communication of heat from the surface of the soil to the atmos- 
phere. It is obvious that a loss of heat from this surface to the 
colder parts of the atmosphere is, to a certain extent, compen- 
sated by a contribution of heat from the hotter parts. There 
must be a continuous interchange of heat between the soil and a 
certain quantity of the atmosphere, of which we know neither the 
height nor the distribution of temperatures within it. To simplify 
the question, we suppose that there is no direct radiation from the 
soil into space ; but we are not necessarily bound to this assump- 
tion, because it is obvious that any amount of heat supposed to be 
radiated into space may just as well be assumed to be radiated 
against a further imaginary layer of the atmosphere, so long as 
we can dispose of the density and temperature of this layer. 
Now, let us take a certain element of mass, dm , of the atmosphere 
at any distance from the soil. Let be the coefficient of 
radiation of the soil against this element, t k its temperature, and 
t' the temperature of the earth’s surface. We know, from 
Newton’s law, that the quantity of heat radiated during the 
infinitely small space of time, dz , from the soil towards the element, 
is given by the formula : 
dv — - (j>{\)dm{t' - t^)dz * 
* This formula obviously includes the case of conduction between the soil 
and the first elements of the atmosphere too — provided we make no 
assumptions on the nature of the function <p( A). If, in what follows, we 
simply speak of radiation between soil and air, we always tacitly refer to 
the probable existence of such conduction. 
