202 THE MECHANICS OP THE EARTH'S ATMOSPHERE. 



The division by T was necessary in this case only in order to give the 

 left-hand side of the equation the form of a difference of entropy. 

 With the help of the equation of elasticity and the equation (la) we can 

 eliminate v and r, ami introduce instead of them the pressure p. The 

 equation then shows us how the quantity a of ice that is formed varies 

 with the chaDge of pressure. The details of this process however in- 

 terest us less than the limits within which it takes place. Therefore we 

 let the subscript index figure refer to the condition in which the mix- 

 ture just reaches the temperature 0° in which therefore ice is not pres- 

 ent, and where o" () =0. On the other hand we let the subscript index 

 figure 1 refer to the condition in which the last particles of water are 

 freezing, in which therefore the temperature just begins to fall below 

 zero. In this condition, evidently, <?=/<- r, since only ice and vapor 

 are now present. If now we substitute these values after introducing 

 the pressure, there results 



mnm p —e , , R e r+q , R e r q _ ft ,«v 



A AR log F —- +* w • - — - • -tjt— X 'j> • zr—2 ' T~ h yjr ( '' 



This equation connects the pressures p and jp„ at which respectively 

 the third stage is attained and relinquished. 



It was not necessary to append an index figure to the quantities e and 

 T since they are alike for the initial and final conditions. 



Fourth stage.— If now the temperature sinks lower, we have then only 

 vapor and ice. The relations that we have to consider are the same as 

 in the second stage, and the final formula is also the same. Only here 

 the specific heat of evaporation has another value from that there given. 

 Here, namely, it is equal to r+q since the heat that is necessary to 

 immediately change ice into vapor must exactly equal the heat that is 

 needed to first melt the ice and then chauge the water into vapor. If 

 we would be perfectly rigorous we ought not to assume q as constant, 

 but must consider it as slightly variable with the temperature, but the 

 differences are so small that here they may remain out of consideration. 

 In this fourth stage we may attain to those low temperatures at which 

 the air itself can no longer be considered as a permaueut gas. 



The four stages that we have here distinguished, one can very prop- 

 erly designate as the dry, the rain, the hail, and the snow stage. 



If one is now in a position such that he is obliged to exactly follow 

 the changes that a mixture containing a considerable percentage of 

 water must undergo, then nothing further remains than to abide by these 

 more complicated formulae. In that case one proceeds in the following 

 manner : First we substitute the values of A and fx in all the equations. 

 Then we substitute the quantities p and T for the given initial condi- 

 tion in equation (L). We then consider the resulting equation and the 

 equation (16) as two simultaneous equations with the two unknown 

 quantities p and T. Solving those equations with reference to these 

 quantities, we obtain that condition through which we must go in pass- 



