232 THE MECHANICS OF THE EARTH'S ATMOSPHERE, 



hence we obtain by integration 



AB x a]o S V v 2 +^(T 2 -v 1 )-lx 2 "=0 (21) 



where we assume the integral to be taken throughout the whole stage 

 from the initial value v x that corresponds the entrance into this stage 

 to the final value v 2 that refers to the exit therefrom, and remember 

 that the initial value of x" namely, x\" is equal to under these condi- 

 tions. If however the integral extends only up to a value of v lying 

 between these two limits and which v can then be considered as vari- 

 able, then the equation can be again brought into a form analogous to 

 that above given and we obtain 



ARia\ogv+ r,,( '"r-lx" = C (22) 



This equation allows us to see directly that for increasing values of 

 v that is to say for continued progressive expansion the quantity of 

 hail also steadily increases whereas on the other hand from [equation 

 (18) or] the expression 



dx= %dv 

 alts 



it follows that an evaporation goes hand in hand with the freezing of 

 the water, so that at the end of the hail stage the quantity of vapor 

 present is greater than it was at the entrance upon this stage. 



With the help of the above described geometrical presentation we 

 represent these results in the following manner. 



The condition that must exist at the entrance upon the hail stage 

 finds its representation at the termination N' of a straight Hue N N' per- 

 pendicular to the chief plane of coordinates and which rises up above 

 the dew point surface. The length of this straight line is x+x'. It 

 cuts the dew point surface at a point N that is distant from the plane 

 of PV by the quantity x. If now the mixture expands along the 

 isotherm then N rises along the dew-point surface slowly upwards, 

 while the foot N of the straight line advances along an equilateral 

 hyperbola. But at the same time, the total quantity x+x' diminishes 

 in consequence of the discharge of the ice and N' sinks correspond- 

 ingly down until N and N' coincide in a single point N 2 and with this 

 the had stage has reached its end. 



It is now of especial importance to learn how much water is thrown 

 down in the form of hail ; this question is answered by the following con- 

 sideration. At the beginning of this stage we have only water and 

 vapor, at the end only ice and vapor while the sum of these in the first 

 and in the second case remain the same, if we take the precipitated ice 

 also into the computation. Let x{ be the quantity of liquid water present 



