378 
-MR. LOUIS UESSOT KING ON THE SCATTERING AND 
§ 2. On. the General Integral Equation for the Scattered Radiation. 
Suppose a mass of gas to be completely enclosed by a boundary 2 and illuminated 
by a distribution of radiation whose intensity at a point (x, y, z) is E, and whose 
direction is defined at that point. Each element of volume will scatter a certain 
proportion of the radiation incident upon it, so that each element besides being 
illuminated by the incident radiation is also subject to the aggregate radiation from 
all the other elements within the surface 2, i.e., to the effect of ,self illumination. 
This constitutes what Schuster* has called the problem of “ Radiation through a 
Foggy Atmosphere.” The problem does not lend itself easily to a complete formula¬ 
tion in terms of differential equations, but can be expressed in terms of an integral 
equation. 
At an element of volume at {x, y, z) the incident radiation is E. At a distance 
r from h' in a direction B with the incident radiation the scattered radiation crossing 
unit area in unit time is denoted by I {'?■, d) Iv r~‘^. If we wish to express the 
dependence of I (r, d) on the position of Sv in the surface 2 we write it in the 
form I {x, y, z, r, d) Sv r~'^. 
Our first problem will be to express the scattered radiation I (r, d) in terms of the 
scattered radiation 1(0, d) per unit solid angle in the neighbourhood Iv. 
Consider the radiation I {r, d) Iv 8o> coming from an element of volume and 
contained in a small solid angle dw in a direction d with the incident radiation. The 
intensity of tlie radiation crossing a spherical surface of radius r is I d) Iv ; that 
cutting the surface r + Ir is 
We thus obtain tlie equation 
— — I ( 7 *, d) Sv Sm = al d) Sv E Sw S'}’+ d) 7’“^ Sv E Sw Sr. . . (lO) 
S?' 
The first term on the right-hand side represents by (8) the energy lost to the element 
of volume 7-^ Sw Sr by the conversion of radiant energy into a rise of temperature or 
into long-wave heat radiation, which transformation we assume to go on at a 
constant rate. 
The second term accounts for the energy lost to the element of volume 7’^ Sw Sr by 
scattering. We neglect the effect of self-illumination within the small solid angle Sa> 
since this only enters into the equation to the second order of small quantities. 
* ScHU.STER, “Radiation through a Foggy Atmosphere,” ‘ Astrophysical Journal,’ XXI, January, 1905, 
p. 1. A somewhat similar problem had been previously considered by the same v'riter in a paper “ The 
Influence of Radiation on the Transmission of Heat,” ‘Phil. Mag.,’ February, 1903. Recently it has been 
shown by Jackson, W. H. (‘Bull. Am. Soc. Math.,’ XVI, June, 1910, p. 4731, that the generalized 
differential equation obtained in Schuster’s paper can be transformed into an integral equation of the 
Fredholm type. 
{l (r. e) 
{r, d) Sr [■ Sv S< 
3oj. 
