312 CLAUSIUS ON THE CONSTITUENTS OF THE ATMOSPHERE 



gard the progress as contitmous through the length of the path, 

 by which we secure the advantage of being able to apply the 

 differential calculus. 



Let ds be an element of the path, and the intensity of the 

 light at the commencement = J, then the loss of intensity en- 

 dured by the light in passing over the distance ds, is denoted by 



J . h ds, 



and h is the constant which expresses the reflective power of 

 the atmosphere. 



The determination of the second constant requires a some- 

 what more tedious development. By means of each ray which 

 descending through the atmosphere meets the eye, a point in 

 the dome of heaven appears to us illuminated ; the direction of 

 the ray may be determined by the point in the firmament from 

 which it appears to come, and its intensity by the brightness im- 

 parted to this point. To denote the different points of the 

 hemispherical dome of the firmament, let two great circles AX 

 and AY (Plate V. fig. 1) be supposed to pass through any one of 

 them, A. On account of the smallness of the arcs which enter 

 into consideration here, we may regard the portions of the coils as 

 straight lines, and make use of them as rectangular coordinates. 



Let us suppose the observer at any point within the atmo- 

 sphere, regarding from this point the portion of the firmament 

 in the neighbourhood of A. At its different points this may 

 appear of different brightness, but will not exhibit the sudden 

 change exhibited by a star upon the dark blue ground behind 

 it ; a gradual increase or decrease of light from point to point 

 will, on the contrary, be observed. For the present purpose, 

 let us assume more definitely, that the brightness v is the same 

 in all lines, such as CD parallel to AY, but that with increasing 

 a?, on the contrary, it diminishes according to the equation 



v = a—x, (2) 



where a is a constant. 



Let us suppose the position of the observer in the atmosphere 

 moved a certain distance backwards, and let him regard the 

 firmament from this new point as before, the light in its passage 

 from his first to his second position having undergone a certain 

 dispersion. By this the rays appear as if they came from other 



