316 CLAUSIUS ON THE CONSTITUENTS OF THE ATMOSPHERE 



obtain a value of \ different from 0, we must have /= co ; but in 

 the calculation we may neglect p in comparison with all distances 

 which can be subjected to observation. We therefore assume 

 that the star observed without the atmosphere is a bright point 

 possessing the intensity \ ; and the question now is, How must 

 it appear after the rays have passed through the atmosphere? 

 Let us return to equation (6), 



^ _ (^-a)2+(y-/3)2 



s 

 Taking the position of the star as the origin of coordinates, when 

 s = 0, for all values of x and y, with the exception of zero, u must 

 disappear. As, however, for or = a andy=^ the above expression 

 does not disappear, then we must have 



and we have therefore 



or setting oc^-^y"^ — r 



A _£!±r 



S 



A ^ 



= -e 4/fc* (9) 



The arbitrary constant A, which still appears in this equation, 

 can be determined in the following manner. The original lumi- 

 nous intensity of the star was X. Now as by the above u is the 

 expression of the brightness which would be observed if the 

 light suffered no loss in its passage through the atmosphere, but 

 was merely changed in direction, then the light of the star must, 

 as regards quantity, be invariable, and only as regards its distri- 

 bution in the firmament be dependent on s. If, therefore, by 

 means of the foregoing expression, we determine by integration 

 the intensity for any value whatever of s, we must always obtain 

 the original value \. We have therefore 

 /ICC r* A -— 



Jo Jo * 



from which it follows that A= -j-, so that equation (9) becomes 



u=-.^e~^s (9./) 



