TRANSMISSION THROUGH AN ABSORBING MEDIUM. 207 



The relationship of the density at any point to the 

 pressure is given by the equation 



In the previous cases the ray has been supposed to pro- 

 ceed through the atmosphere from the earth ; if the path 

 of the ray is through the atmosphere towards the earth, 

 the formulae will require a little alteration. 



Let h be the distance of the luminous object from the 

 earth, and I its brightness at that distance, 6 its distance 

 from any layer of the atmosphere below it, and t the 

 distance of that layer from the earth. Then we have 



Hence the expression for the density becomes 



d=T)e a , 



and the formula for the intensity of the transmitted light 

 becomes 



I=I/ v a <- a >. 



If, in this formula, we make = h, and then make h 

 large, we have for the limiting value of the intensity 



I = I Q e *. 



In equation (7) the light is supposed to proceed through 

 the atmosphere vertically. When this is not the case the 

 expression for the intensity becomes more complex. In 

 treatises on geometrical optics may be found the following 

 differential equation of the path of a ray of light which 

 has passed through a medium, of which the refractive 



