324 CLAUSIUS ON THE CONSTITUENTS OF THE ATMOSPHERE 



such a large quantity of reflected light ; and the assumption of 

 an absorption which goes on at the same time could therefore 

 only cause a change in the numerical values, which would by no 

 means suffice to compensate the differences which show them- 

 selves further on. 



By means of this value of h, we obtain from the foregoing 

 table the following values of k : — 



1-0003 I 1-00003 I 1-00001 I 1-000001 I 1-0000001 I 1-00000001 

 0-00099 I 0-00013 I 0-000049 I 0-0000061 I 0-00000072 I 000000084 



These are the values which must be set singly in equation (10«), 

 which latter however w^e will first simplify by assuming that the 

 star is in the zenith, by which s is rendered =1, and (10«) 

 becomes 



v=ve ^* (20) 



The expansion of the luminous circle, which we see in the place 

 of the star, depends now only upon k, and moreover so, that the 

 smaller k is, the more quickly will the brightness diminish as 

 r increases. To furnish a conception of the speed of this dimi- 

 nution in the following table for the different values of n, and 

 hence for those of k, those magnitudes of the radius r are stated 

 which correspond to a brightness equal to y^th of that at the 

 centre : — 



1-0003 I 1-00003 I 1-00001 I 1-000001 I 1-0000001 I 1-00000001 



5^29' 2° ri3' 25'-7 8'-8 3'-01 



Calling to mind that the radius of the sun's disc is about 16', 

 we see that if the reflexion were caused by the frequent passage 

 from vacuum into the masses of air, or from nitrogen to oxygen, 

 and vice versa, then a star, in case it possessed sufficient intensity 

 to be at all visible, would appear far greater than the sun. In 

 the case of the other smaller refractive indices, the magnitude 

 of this luminous circle certainly diminishes, but in all cases the 

 result remains altogether irreconcileable with the reality ; for a 

 fixed star, even observed through a highly magnifying telescope, 

 appears as a point, and not in the zenith alone, but also near the 

 horizon. 



