270 Proceedings of Royal Society of Edinburgh. [sess. 
where K is a constant depending on the solar radiation as well as 
the thermal capacity of the soil. Eliminating t\ we arrive at the 
differential equation of the second order 
^ + -u)= - hK cos cf) cos 8 sin £ , 
and consequently at the general integral 
, , - 0382 hz 
t — u-k-c^e + c 2 e 
2 G1S// ' + a cos cos 8{(1 - h 2 ) sin £ + 3 h cos £} 
hK 
a ~ 1 + 7h 2 + id ’ 
£ representing the sun’s hour angle, 8 his declination, and <£ the 
latitude of the place of observation. 
Obviously we have to take this integral between the limits - r 
and + r, r being the semidiurnal arc of the sun, because our equa- 
tions can only refer to points within this interval. Besides, it can 
he easily proved that both the arbitrary constants have to dis- 
appear. This may he inferred immediately from what we said 
about the constancy of temperature at constant radiation. Hence 
we conclude that the integral representing the change of tempera- 
ture during the day must have the form : 
t — t + a cos <f> cos 8 
(1 - h 2 ) sin £ + 3 h cos £ 
— T 9 
t- T being the temperature of air at any moment near sunrise. We 
may write this equation in a simpler way by putting 
(1 -li 2 )- a = a sin?; j 
3h • a = a cos v 
t Q = t _ T — a cos cos 8 cos (t + v ) , 
whence we derive 
t = t 0 + a cos cos 8 cos (£ - v) 
1 - h 2 
The angle v being determined by the relation tang v — — , 
the time of maximum of temperature can be found by introducing 
the above given value h = 0‘375 into this equation. We find 
v~37°’4:~ 2 h 29 m p.m. ; a value which agrees perfectly with the 
well-known average epoch of maximum for continental climates 
derived directly from observations* 
