ABSORPTION OF LIGHT IN GASEOUS MEDIA. 
389 
is considered in a formula obtained by Rayleigh.'^ The term y in C = c + y allows 
for attenuation by absorption alone (he., without scattering). The second terms in 
(59) and (6O) represent the contribution of self-illumination to the scattered radiation 
coming from any particular direction. An evaluation of this effect has not, so far as 
the writer is aware, been submitted to calculation although the importance of the 
effect is realized both by Kelvin t and Rayleigh,* and in an analogous problem by 
Lommel.| 
The expressions (59) and (6O) for the scattered radiation from any direction besides 
depending on the coefficients of absorption for the radiation of wave-length under 
consideration, depend also on the angular co-ordinates of direction 0 and S'® well as 
on ^ the zenith distance of the sun. If we consider the intensity from zenith sky tlie 
expressions are greatly simplified. Writing <p = 0 and 6 = f, (59) gives, 
T (0, 0 = 
iO a 
K 
Ce-^G{C(sec f-l)}+i 
(C, 0) 
(64) 
where <I> (C, 0) is given by (57) and is tabulated in Table V. 
It is not difficult to construct a double-entry table in terms of C and i giving the 
values for the functions which occur in (64) so that observations on zenith sky are 
most appropriate for comparison with the results of calculation from the attenuation 
coefficients determined by observations at the same time. 
The intensity of sky radiation from the direction of the horizon (0 = -) is given by 
T(|.f 
Mu 
(e)S, 
-C sec? 
Kn 
(65) 
where by (62) 6 is given by cos 9 = sin f cos <p and ffi(C, |-7r) is defined by (58). It 
will be noticed that this formula unlike the approximate one (63), obtained by Kelvin, 
* Rayleigh, ‘Phil. Mag.,’ XLL, p. 116. 
t Kelvin, loc. cit., p. 302, sect. 54. 
J Lommel, E., ‘ Sitzungb. der math.-phys. Classe der K. Bayer. Akad. der Wiss.,’ Bet 17 (1887), 
p. 95; (the analysis is reproduced by Muller, ‘ Photometrie der Gestiriie,’ Leipzig, 1897, pp. 47-52). 
In working out the scattering of radiation by opaque, diffusely reflecting surfaces on an absorption 
theory, Lommel obtains a first order approximation to the effect of self-illumination in giving rise to 
deviations from Lambert’s Law of diffuse reflection, the modified formula being referred to as the 
Lommel-Seeliger Law of Illumination. This-problem is included as a particular case of the investigation 
of the present paper. Lommel’s formula for diffuse reflection is represented in the present instance by 
equation (69), which is the solution of an integral equation expressing exactly the effect of self-illumination. 
Muller points out that even Lommel’s modification of Lambert’s Law does not represent exactly the 
results of observation on diffuse reflection; it may happen that the more complete solution represented bj" 
(69), adapted to the case of intense absorption, gives a better representation of fact. This point must, 
however, be left over for further investigation. 
