WHICH REFLECT LIGHT. 309 



Instead of these considerably divergent nunibers let us take the 

 mean of both, the loss endured by the light will therefore be 

 0*25 . Now if this loss were brought about by reflexion on spheres 

 of water, then by comparison with the number above found, 0*12, 

 we must infer that so many spheres are present that the average 

 number crossed by every ray must be two. From this it follows, 

 further, that when the sun is not in the zenith, but in the 

 horizon, where the path of the light through the atmosphere is 

 thirty-five times larger, then the number of spheres crossed by 

 each ray on an average must be 70. 



In the estimation of the dispersion caused by these spheres, 

 we will regard the path of the light through the atmosphere so 

 long merely, that each ray cuts one sphere on the average. 



To simplify the investigation, let us for the present assume, 

 not that the average number crossed by every ray is one, but 

 that every ray cuts one sphere exactly. The total light, after 

 passing through this distance, must be dispersed in the same 

 manner as that which has passed through the single sphere. 

 This dispersion is easily determined by means of the ordinary 

 law of refraction, Fresnel's formula of reflexion being made use 

 of and the reflected light being subtracted. The sheaf of light 

 which falls parallel upon the sphere will diverge, after its passage, 

 nearly conically, but so that the intensity of the light in the 

 axis of the cone, which constitutes the production of the inci- 

 dent rays, is greatest, and diminishes the further the light is 

 deflected from its original direction. Denoting the intensity in 

 the axis of the cone by 1, then for the intensities J, which cor- 

 respond to the various angles of deflection 7, we have the fol- 

 lowing numbers : — 



y I I 5» I IQO I 200 I SQO | 40" | SQO | 6OO 

 J I 1 I 0-98 I 0-83 1 0-54 | 0-28 | 014 | 0-06 | 0-03 



Under this assumption, therefore, instead of observing the 

 sharply defined disc of the sun in the zenith on a comparatively 

 dark ground, we should see a large bright space extending with 

 gradually diminishing brightness from the zenith downwards to 

 beyond 60°. 



We must now free ourselves from the foregoing limitation, 

 that each ray meets exactly one sphere in its passage, and in- 

 vestigate how the result is modified when we simply regard 1 



