320 CLAUSIIJS ON THE CONSTITUENTS OP THE ATMOSPHERE 



that each ray encounters on the way a^ not one sphere on the ave- 

 rage, but exactly onQ sphere. In the former considerations respect- 

 ing the spheres of water, attention has been drawn to the differ- 

 ence between these two assumptions. Here, however, where the 

 light does not reach the eye after it has passed over the distance 

 a, but on the contrary has to travel over many such distances, 

 it is not of importance to inquire in each particular case into the 

 change of the form of the dispersion, but into its magnitude ; 

 and this is, as may be readily seen, the same in both cases. The 

 diminution suffered by the ray which encounters no sphere, and 

 therefore remains undispersed, will be compensated by that suf- 

 fered by others which meet more than a single sphere, and is 

 thus repeatedly dispersed, and, as in the case of reflexion, the 

 smaller the refractive face of the masses the more perfect this 

 compensation w^ill be. We can, therefore, without further hesi- 

 tation, make the assumption, that on the distance a each ray 

 encounters one sphere exactly. 



According to this, the length a acts upon the total quantity 

 of light as a single sphere on the light which passes through it. 

 As before, in the case of reflexion, we fixed our attention on a 

 single sphere, so here also let us again suppose the sphere divided 

 into its elementary zones. The portion of light that falls on and 

 penetrates these suffers a definite dispersion, and it must be in- 

 vestigated how much luminous intensity passes through EF 

 when this portion of the light only is taken into account. From 

 this we can afterwards find by integration the total intensity 

 which passes through EF. 



Assuming on the firmament any element of surface d(o to the 

 left of CD (Plate V. fig. 2), by the point M, with the coordinates 

 X andy; this element possesses the intensity {a — x)d(o, and 

 the light proceeding from it, which falls upon the elementary 

 zone in question, is 



= [a—x)da),2&\nico^idi (13) 



Of this, according to the foregoing, the quantity 



N^ «• . .^. r, lsin2(i-i') 1 tan2(i-i')-)2 

 [a—x)d(o.2smico^idi\ 1— - . ^.. .,. — -~ — ^). , ./ 

 ^ ^ L 2 sin^ {t + ^') 2 tan^ {t + v)J 



passes through the sphere, and then, instead of pursuing its 

 former direction, is by the two-fold refraction deflected from it 



