60 
Proceedings of the Koyal Society of Edinburgh. [Sess. 
slowness of the processes concerned. But in the case of (1) movement in 
latitude, (2) the formation of extensive cloud-screens, (3) changes of 
diathermancy due to the formation of thin haze, etc., we perceive causes 
entirely adequate to produce gradual but appreciable changes of temperature 
(say, variations of 10-15 degrees absolute) in extensive masses of air. 
The next step is to investigate the result of such changes occurring in a 
layer of air possessing considerable horizontal extension (say, from 10,000 
to 500,000 square miles). 
The second law of dynamics gives us the well-known equation of 
vertical equilibrium, 
dp 
dE 
(s+ a )p, 
where H is the height above some fixed level, p and p the pressure and 
density of the air at that height, g the acceleration of a freely falling body 
in vacuo at that height, and a the vertical acceleration of the air. 
We are concerned here with slow, cumulative changes continuing 
through many hours or even days. In considering the equilibrium of 
extensive masses of air for such periods of time, the mean value of a must 
be regarded as negligible in comparison with g. For if we supposed it to 
remain even for fifteen minutes equal to ]_qq^qqq result would be a 
vertical current exceeding 80 m.p.h. It is doubtful if such vertical speeds 
are achieved even in the most violent thunderstorms, and they are certainly 
not long maintained, as the air has considerable vertical stability owing to 
the low average lapse rate. Thus, a may safely be neglected in comparison 
with g, giving the simpler equation 
(1) 
where g is almost a constant, or, more accurately, a function of H. 
For moderate changes which are here dealt with, unsaturated air 
behaves approximately like a perfect gas. Thus the following equations 
may be assumed, for the first treatment : — 
^ — (Charles’s law), 
E y 
p = e^e^i (^ aw adiabatic expansion), 
where k is the volume which unit mass of air would have at unit 
temperature and pressure if it continued to behave like a perfect gas 
under these conditions, 0 is the temperature on the absolute scale, e is 
the base of “ natural ” logarithms, E is the excess of entropy of unit 
mass of air at pressure p and temperature 0 over its entropy at unit 
