WHICH REFLECT LIGHT. 313 



points, so that the intensity of the light in the firmament ap- 

 pears differently distributed. Now, as in the foregoing case, 

 the spaces left of CD all possessed a greater intensity than the 

 spaces to the right, the dispersion, the tendency of which is 

 to compensate these differences, will cause a portion of the 

 light to be apparently transferred from the left to the right of 

 CD, and moreover an equal quantity in all points of CD, 

 because, as regards y, the brightness is constant. We can 

 therefore express the dispersive power of the atmosphere by 

 stating how much of the luminous intensity passes through an 

 element =dy of the line CD from left to right, while the rays 

 have traversed the element ds through the atmosphere. De- 

 noting this quantity by 



k dy dSf 



then k is the second constant sought. 



By means of these constants, h and k, we can form the differ- 

 ential equations by which the changeable distribution of the 

 luminous intensity in the firmament will be determined. For this 

 purpose let us fix our attention on the point M (Plate V. fig. 1) 

 on the firmament, the coordinates of which shall be x and y, 

 and let us take the element MN MJW—dxdy. Let the bright- 

 ness of this element, as seen from any point of the atmosphere, 

 be V. Removing the point of observation, the distance ds, back- 

 wards in the direction of the rays, the element regarded from 

 this new point appears no more of the brightness v, for this has 

 been changed both by reflexion and dispersion. By the former 

 of the intensities, v dx dy, the portion 



h,v dxdy ds (I.) 



is lost upon the way ds. To ascertain the effect of the latter, it 



must in the first place be remarked, that the above expression, 



kdyds, which resulted from the assumption of the definite 



distribution of the intensity expressed by equation (2), may be 



easily changed so as to be applicable to every other mode of 



dv 

 distribution. According to equation (2), we have — = — 1, and 



it is thus manifest, that, to introduce the generalization just 



dv 

 mentioned, it is only necessary to set -j- in the place of —1. 



SCIEN. MEM.-^Nat. Phil. Vol. I. Part IV. Z 



