298 Proceedings of Royal Society of Edinburgh. [sess. 
mission varies greatly with the wave-length and according to a 
law which experiment alone can discover, the use of a mean value 
of a for the whole radiation will necessarily give too great a value 
for the transmissibility through increasing masses of air. Bearing 
this in mind, we may for the present purpose assume the law 
mentioned, although we know that it is only a first rough approxi- 
mation and will give too high a value for the transmissibility when 
the altitude of the sun is small. 
Langley’s broad result is that the energy of the solar radiation, 
which reaches the earth’s surface after transmission through the 
vertical depth of atmosphere, is about two- thirds of the energy 
which would reach the surface if the air were absent. Calling this 
coefficient of transmission a , we see that if £ represents the zenith 
distance of the sun the mass of air traversed is roughly propor- 
tional to sec £. The radiation falling normally on unit surface is 
therefore proportional to a sec C. Hence the radiation falling on 
each square centimetre of the earth’s horizontal surface is propor- 
tional to cos £. a sec C. If we multiply this by the element of time 
and integrate from sunrise to culmination, we shall get half the 
quantity of solar energy which falls on each square centimetre of 
the earth’s surface during one day. Let A be the latitude of the 
place and 8 the sun’s declination at the time considered, then the 
zenith distance £ is connected with the time by means of the 
formula 
cos £ = sin A sin 8 4- cos A cos 8 cos wt 
where w is the angular velocity of the earth about its axis. 
The evaluation of the integral 
can be effected with sufficient accuracy by graphical methods. 
To this end the quantity cos £. a seG ^ was calculated for a series 
of convenient values of £, and then, by means of the formula given 
above, the corresponding values of t were calculated for the posi- 
tions of the sun at intervals of a month, ranging from summer to 
winter solstice. For each value of the sun’s declination a curve 
was then drawn, the abscissae of which were the times reckoned 
from culmination, and the ordinates the corresponding values of 
