310 CLAUSIUS ON THE CONSTITUENTS OF THE ATMOSPHERE 



as the average number. It is therefore assumed that the 

 number of spheres present is exactly sufficient^ provided the 

 arrangement were made so to block up the passage of the rays 

 that each ray must pass through a sphere ; these spheres, how- 

 ever, being now supposed to be arbitrarily scattered through 

 the atmosphere. In this case, of course, many rays would pass 

 through quite free, and others, on the contrary, meet with more 

 than one sphere, and, according to the principles of probability, 

 we can determine the ratios of the various portions of the light 

 which meets one, which two, three, four, &c. spheres. Let A 

 be the total quantity of light, and e the basis of the hyperbolic 

 logarithms, that is, 



e=H-[ + ^ + ^H- =2-71828 . . . 



we then obtain the following values : — 



Quantity of light which meets no sphere =— =A.03679 



e 



A 1 



one „ =— •— =A-0-3679 



two „ =--i7=A.0-1839 



e 2.' 



A 1 



three „ =— •-7=A.O-0613 



(1) 



e3. 



According to this, one-third of the light coming from the 

 sun would, it is true, preserve its parallelism, but a second 

 equal quantity would exhibit the dispersion given above, while 

 the remaining portion would suffer a still greater dispersion. 



Such dispersion does not at all exist, and still the amount is 

 made much too small, as of the shortest possible path of the 

 light we have only taken the half. It is therefore plain that 

 the assumption of the reflexion being caused by solid spheres 

 of water suspended in the atmosphere must be altogether re- 

 jected. 



Instead of water, let us suppose the case of masses pos- 

 sessing smaller refractive force, and whose number compensate 

 for their want in this particular, we thus obtain, it must be 

 admitted, a feebler dispersion than before. As long however as 

 the refractive power remains in any degree considerable, we 

 may always convince ourselves by considerations like the fore- 



