WHICH REFLECT LIGHT. 315 



the distance s through the atmosphere, neglecting the reflexion, 

 and taking the dispersion alone into account. To determine 

 from this quantity u the true brightness v, we have only to mul- 

 tiply it by e~'''\ 



The equation (5) must now be integrated, and moreover the 

 expression which we seek must be so characterized, that it not 

 only satisfies the differential equation (5), but also for 5=0 it 

 shall exactly represent the appearance which the firmament 

 would present to an observer without the atmosphere. The first 

 condition will be fulfilled by the expression, and by every sum 

 of expressions of the form 



^ (.r-a)^+(y-/3)^ 

 U=—€ 4/t* ^ (6) 



where A, a, and ^ are arbitrary constants. In regard to the 

 second condition, the expression which fulfils it quite generally 

 shall be only briefly stated here, as for the present purpose the 

 examination of a particular case is sufficient. Let u=F{x, y) 

 be the equation which denotes the original distribution of the 

 luminous intensity on the firmament, then, as Fourier has shown 

 in the theory of heat, we have 



dP¥{a,P)^^e ^ks- ; . (7) 



1 C+ * /^+ ' 

 -\ da\ 



an expression which satisfies the differential equation (5), and 

 for 5=0 passes over into w=F(.r, y). 



As a particular case, the examination of which we must now 

 enter upon more closely, we will choose the appearance of a 

 single fixed star. Without the atmosphere, a certain portion of 

 the firmament w^ould appear completely dark, and only a small 

 circular space would shine upon it with considerable brightness. 

 Let the radius of the little circle =p; hence the space taken 

 up by the star =/oV; and let us further denote the brightness 

 of this space by t ; then the total luminous intensity (X,) of the 

 star will be expressed by 



\=p^7rt (8) 



The space pV is however with all fixed stars so small, that they 

 appear to us as points. Strictly speaking, nevertheless, we 

 cannot set p = 0, for then, in the foregoing equation, in order to 



z 2 



