134 THE ROYAL SOCIETY OF CANADA 
Atmosphere” draws attention to the importance of the effect in astro- 
physical problems and obtains a representation of the effect by means of 
differential equations; the results are applied to problems of the re- 
versal of lines in certain steller spectra to be attributed to the effect of 
the atmospheres of these stars. 
The application of the theory of integral equations to problems of 
this type seems to be a new result and opens out a method of attack- 
ing a great number of problems of the type just considered. 
2. An immediate application of the general equations just dis- 
cussed can be made to the problem of absorption and scattering of solar 
radiation by the earth’s atmosphere. In this case the general integral 
equation is reduced to one in a single variable. By means of a perfectly 
general transformation of variables the problem is reduced to one of a 
homogeneous atmosphere contained between two parallel planes at a 
distance H apart. This transformation is independent of any law of 
density or temperature gradient in the atmosphere and only requires 
that the planes parallel to the earth’s surface be planes of equal density. 
The integral equation for the scattered radiation is then derived, 
and by a consideration of the rate of accumulation of energy in an 
element of volume, the result is shown to be consistent with the well- 
known law of attenuation. 
—K,(Hax-x) sec Z ® (vi) 
E (x) —196 
E (x) is the intensity of solar radiation per unit area normal to the 
: Ia vaxr 7 . Ja aT r >, Coy No / 
sun’s rays at a height x above the earth’s surface where De fe p/ po dx, P 
being the density of the atmosphere at a height x and p, that at the 
earth’s surface. K, is the coefficient of attenuation under conditions 
of standard temperature and pressure. 
D] 
3. Progress towards the solution of the integral equation requires 
the development of a special method of approximation. It is shown in 
general that the solution of an integral equation of the type (iv) must lie 
between two limits called the extreme solutions; a solution lying between 
these is derived and called the mean solution: it represents a value 
probably not far from the correct one in the applications considered. 
4. The approximate solution of the integral equation leads to a 
number of transcendental functions which recur so frequently in the 
course of the analysis that they are designated by a special notation 
and tabulated: 
