202 THE MECHANICS OF THE EARTh's ATMOSPHERE. 



The (livi.siou by T wns uecessarv in this case only in order to give tbe 

 left-hand vside of tbe equation tbe form of a dilferenee of entropy, 

 ^Vitb the hel]) of the equation of elasticity and the equation (1^0 wecau 

 eliminate v and ;', and introduce instead of them the pressure p. The 

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

 with tbe change of pressure. Tbe details of this process however in- 

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

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

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

 ent, and where (7ii=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 wbicli therefore tbe temperature just begins to fall below 

 zero, in this condition, evidently, (T=// — i', since only ice and vapor 

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

 the pressure, there results 



A AR log +A ^ . . — ^ — A.^ . . „— /^ y~ =U . . ,3). 



This equation connects the pressures ^^o and^Ji, at which respectively 

 the third stage is attained and relin(]uished. 



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

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



Fourth stage. — If no^v the temperature sinks lower, we have then only 

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

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

 the specific heat of evai)oration has another value from that there given. 

 Here, namel}', it is equal to r-{-q since the heat that is necessary' to 

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

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

 we would be perfectly rigorous we ought not to assume q a-< constant, 

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

 difl'erences are so small that here they may remain out of consideration. 

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

 tbe air itself can no longer be considered as a permanent gas. 



The four stages that we have here distinguished, one can very i)rop- 

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



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

 tbe changes that a mixture containing a considerable percentage of 

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

 more complicated formulne. In that case one i)roceeds in tbe following 

 manner : First we substitute tbe values of A and // in all tbe equations. 

 Then we substitute tbe quantities jj,, and To for the given initial condi- 

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

 equation {lb) as two simultaneous equations with tbe two unknown 

 quantities j) and T. Solving those equations with reference to these 

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



