ABSORPTION OF LIGHT IN GASEOUS MEDIA. 
383 
radiation in tlie earth’s atmosphere reduces to the case of an atmosphere of 
uniform density contained between two parallel planes X = 0 and X = H. 
This transformation is independent of any law of density with height, 
provided the planes of equal densitj'' are parallel to the earth’s surface. 
§ 4. On the Approximate Solution of Integral Equations. 
The integral equation ( 2 l) is of the Fredholm type,* 
w (.r) =/(»;)+[ M (^) K (x, c?5..(24) 
Except for special forms of the kernel K {x, i) the formal solutions of (24) do not 
lend themselves easily to numerical evaluation : we therefore develop a method of 
approximation which applies with sufficient accuracy to the problem in hand. 
Suppose for all values of x between x-^ and x.^ that f{x) lies between A and a (A>a). 
Then to a first approximation 
pi , 
n (a;) lies between A + A K (a;, f) and a + a\ K (a:, ^) 
*' Xl I] 
provided K {x, is everywhere positive. 
We write 
^(a:) = I K(a;, ^)c?^,.(25) 
then if for all values of x, cp {x) lies between B and h (B>6) we have to a second 
approximation 
a + {a + ah) h<.u (x)<A + (A + AB) B, 
or 
a{ \ -\-h-\-E)<ku (a;)<A (l + B + B^). 
A repetition of the process shows that 
r^(l+h + /fi+/d+...)<H(a;)<A(l+B + B^ + B^+...). 
If IBI < I both series are convergent and 
ei<n(x)<e 2 where e-^—ajl—h and €3 = A/l—B. . . . (26) 
Substituting in (24) we see that the solution u (x) lies between the limits 
U]^{x) =f{x) + ei<p{x), and Ug (x) =/(a;) (a:).(27) 
(x) and U 2 {x) may be called the extreme solutions. 
* Cf. Bocher, M., ‘An Introduction to the Study of Integral Equations,’ Cambridge, 1909, p. 14; 
also Bateman, H., “ Report on the History and Present State of the Theory of Integral Equations,” 
‘ Brit. Assoc. Report,’ 1910, p. 25. 
