WHICH REFLECT LIGHT. 31? 



Finally, to obtain, instead of w, the true brightness v, we have 

 only, as already stated, to multiply by e~**" ; hence we obtain 



Denoting then the brightness at the origin of the coordinates, 

 where r=0, by Vq, we have 



^1 —hs 



and introducing this value, the above equation becomes 



v = v^y.e 4/5;* (10«) 



From this formula we find, that after the rays have passed 

 through the atmosphere, instead of a very bright point, a cir- 

 cular space must be seen, the brightness of which diminishes 

 from the centre outwards. The nature of this decrease, however, 

 whether it be slow or sudden, depends upon the magnitude of 

 the constant k. The meaning of this constant being already 

 fixed, the next point is to determine its numerical value, which of 

 course will be different according as the refractive index chosen 

 for the reflecting masses is greater or less. The numerical value 

 of the other constant h may be easily determined from the above- 

 mentioned experiments of Bouguer and Lambert ; and it is 



k 

 therefore only necessary to determine the ratio t in order to 



find the value of k itself. 



It is assumed that in the uniformly dense atmosphere reflecting 

 masses of small refractive power are suspended ; and according 

 to our former discussion, we can, without determining anything 

 regarding their actual shape, consider these masses as spherical. 

 Assuming in this atmosphere a length of path (o-) in which each 

 ray strikes an average of one sphere, and expressing the reflexion 

 and dispersion effected in this distance as functions of the index 

 of refraction (/^), then will the comparison of these two functions 

 give us the sought relation between h and k. 



In the first place, as regards the reflexion, we have only to 

 examine how much of the light which falls parallel upon a single 

 sphere will be reflected by the latter, inasmuch as in the case of 

 the spheres of water above, we may conclude that the same re- 



