90 Proceedings of the Royal Society of Edinburgh. [Sess. 
where 
a 2 = \o? — b, /3 = ^a — a, y = ^a + a 
^t = c o + c \ e ~^ + c 2 e ~ yt 
X t = e^(2a)" 1 F(2) and X 2 = e7^(2a)- 1 F(^) 
T 1 = e~ frffXflx and T 2 = e ~ yffX^dx. 
(2) Derivation of the Differential Equation (I ). — I have adopted for 
the night -temperature curves the differential equations proposed by 
Weilenmann. These equations are reproduced in Dr Halm’s memoir on 
the daily temperature curve.* For the temperatures at day time Dr Halm 
has added to these equations a term which represents solar radiation. I 
also have introduced solar terms, but I have chosen another expression for 
them and have treated the problem in a different way. 
We have to represent the average temperature curve, and may suppose 
that it belongs to an average state of atmosphere characterised by a certain 
degree of transparency. In such an average state the transparency is less 
than that of a clear sky, and the radiation of the soil and lower strata will 
be less than in that ideal case. I assume with Weilenmann as a working 
hypothesis that radiation takes place towards and from a layer in the 
upper strata of the atmosphere where the temperature, r m , remains constant 
during the day, though it may change in the course of the year. Let t 
designate the temperature of the soil in the neighbourhood of the ther- 
mometer. The amount of heat gained on unit area of soil by radiation and 
also the corresponding rise in temperature of the soil are nearly proportional 
to (t w — t). Heat is also transferred between these two places by convection^ 
and the corresponding increase is given by a similar term. The correspond- 
ing gain in the temperature of the soil thus becomes 2a m (r m — r')- The 
soil also gains heat by convection and radiation from the air at the place 
where the thermometer is placed ; the rise in temperature is proportional to 
t — t, r designating the temperature at the thermometer. The rate, r, at 
which the temperature of the soil rises becomes ( — p'r + q'r — m'), apart 
from solar radiation. 
In like manner the rate, r, at which the air temperature changes is 
( — pr + qr— m). p, q, m, p', q', m' are constant quantities. 
These expressions also account for any change in temperature arising 
from radiation or convection from or to a region where the temperature is 
proportional to either t or f. 
The two values of r and f furnish the two equations proposed by 
• Nova Acta der Ksl. Leop. Carol. Deutschen Akademie der Naturforscher, Bd. lxviL 
No. 2, p. 10. 
