384 
ME. LOUIS VESSOT KING ON THE SCATTERING AND 
If now a represent the mean value of ^(a:) between and X 2 , i.e., 
1 
f{x) dx. 
.(28) 
and /3 stand for the mean value of 0 {x) between x-^ and X 2 , i.e.. 
f3 
^ __ 'T* 
t4/9 
0 (x) dx, . 
(29) 
1 ‘' 2-1 
while e = a/(l—/3), the solution u{x) — f {x)-\- efj) {x) may be called the mean solution, 
while the three solutions may be expressed in a single formula by the notation 
« (*) = /(•■*) + ( e ) f (*)■ 
(30) 
69 / 
We notice that (.x) < m (ic) < M 2 (;r). In the applications to be considered, (x) 
and U 2 (x) are sufficiently close to warrant the use of the mean solution ii (x). As 
long as u (x) does not depart far from the arithmetical mean of the extreme solutions, 
^ {mi (x) + M 2 {x)}, we may take the value of u (x) to represent an approximation not 
far removed from the exact solution of the integral equation (24). 
It is perhaps worth noting here that to a higher degree of approximation the 
approximate solution of the integral equation may be written 
U (x) = /(x) + [ f{i) K (x, i) 
^ X\ 
. . (31) 
In the following sections the solutions (31) involve the evaluation of troublesome 
integrals, so that the simpler but somewhat less accurate approximations given in (30) 
will be employed. 
§ 5. On the Solution of the Integral Equation for Sky Radiation. 
In the integral equation (22) we may express to a first approximation the 
dependence of the scattered radiation in any direction on the angle which that 
direction makes with the incident radiation by retaining the term (O) instead of jl^, 
in the fii’st term of the right-hand side of the equation which may be written, on 
making use of (23), 
