PAPER BY DR. HERTZ. 201 



This equation is deduced in Clausius Mech. Warmetheorie, vol. I, sec- 

 tion vi, art. 11 ; and in it c is the specific heat of liquid water, r the 

 external latent heat of vapor, both of them expressed in units of heat. 

 Therefore the total heat communicated to the mixture is 



dQ =X { c.d T+ ART dv J + dT ( -£) + W&T. 



Here also we have to put dQ = 0, then divide by T and integrate. 

 With the help of the equation of elasticity and equation (la) we elimi- 

 nate the quantities v and rfroin the integral equation, and thus obtain 



(xo r + „c~) log^+lAB log^^- 



+4Ai^-%^\=« < 2 >- 



Here also the quantity on the left hand that is equated to zero 

 represents the difference of the entropies between the final and the 

 initial conditions of a kilogram of the mixture. The equation thus 

 obtained can be used until the temperature attains the freezing point, 

 then we arrive at the third stage. 



Third stage. — In this case, in addition to the vapor and the liquid water, 

 the air contains also ice. By further expansion of the air, the temper- 

 ature will now not sink immediately further, for the latent heat of the 

 freezing water will, even without a lowering of temperature, furnish 

 the force necessary for overcoming external pressure. But the heat 

 of liquefaction must not be applied to this purpose only, but also to the 

 evaporation into vapor of a part of the already condensed water. 

 For since the volume increases during the expansion without allowing 

 the temperature to sink, therefore at the end of the process again, 

 more water is become vapor than before, therefore the weight of the 

 ice that is formed will be less than that of the fluid that was present. 



Let now, again, v be that portion of yu that is in the form of aqueous 

 vapor, 6 the part that exists as ice, and q the latent heat of liquefaction 

 of a kilogram of ice. T, e, r are constants. Since therefore dT=0, we 



fill 



have now only to communicate to the air the quantity of heat A ART— 



and to the water that we evaporate the quantity of heat rdr, and to the 

 water that we allow to freeze the quantity —qda. Therefore the quan- 

 tity of heat given to the whole mixture is 



dQ=\ART d -+rdv-qda. 



If we put dQ=0, divide by Tand integrate, there follows 



v 



\ARlog^-+^(r— n)-^{0— <r )=0 



