382 
MR. LOUIS VESSOT KING ON THE SCATTERING AND 
where Ei{—x) is Glaishee’s"^ Exponential Integral denoted by 
Ei{ — x)= — u ^e ''du. 
d X 
(21) 
The integral equation (19) now takes the form 
J(X) = ^,E(X)-2 
TTM,, 
rj(X').£’^ { - K„(X - XO } dX' + rJ(X')^b[ - K„(X' - X)} dX' 
Jo Jx 
( 22 ) 
[The differential equation by means of which E may be expressed as a function of 
X can be obtained by a consideration of the rate of accumulation of energy of the 
direct solar radiation between the planes x and x + dx. An analysis similar to that 
by means of whicli equation (13) was obtained, together with the transformations of 
equation (18), lead to the expression 
E (X) = .(23) 
where S is the intensity of solar radiation outside the earth’s atmosphere corresponding 
to a given wave-length and ^ is the zenith distance of the sun. 
If we consider the rate of accumulation of the total energy (including both direct 
and scattered radiation) between tlie planes x and x + dx, it can be shown by making 
use of the integral equation connecting the direct with the scattered radiation that 
the exponential law of transmission expressed in (23) is valid.!] 
It is well to state clearly the assumptions involved in obtaining (13), (19), and (23):— 
(i) In obtaining the differential equation leading to (13) the direct radiation is 
considered independently of the scattered radiation within the. small solid 
angle w. 
(ii) The integral equation (22) assumes as an approximation that the radiation 
scattered by an element of volume is distributed equally in all directions. 
(iii) With these two conditions it can be shown from a consideration of attenuation 
of total radiation in a thin layer dx that the ordinary exponential law of 
transmission (23) follows ; i.e., that the transmission of direct radiation 
may be considered independently of the scattered radiation. 
(iv) By means of the transformation (18) it is shown that the problem of scattered 
* Glaisher, ‘Phil. Trans.,’ 1870, p. 367. 
[t Xote added Sejitember 20, 1912 .—The calculation referred to was given at length in the paper as 
originally communicated; the analysis is, however, somewhat tedious and hardly necessary in view of the 
fact that (13) is obtained according to the assumption that the direct radiation is considered independently 
of the scattered radiation within the small solid angle w. It is to be expected, therefore, that the 
attenuation of direct radiation in a parallel beam of solar radiation may be considered independently of 
the scattered radiation con.sistently with the above assumption leading immediately to (23); the considera¬ 
tion of the attenuation of the total radiation confirms this point and leads to the same result. The writer 
is indebted to the referees for the above suggestion.] 
