PAPER BY DR. HERTZ. 203 



ing from the first to the second stage. The values thus obtained are 

 then to be substituted as p and T in equation (2). By substituting 

 T=273° in the equation thus obtained, we obtain that p which occurs 

 in the equations of the third stage. If now we further determine from 

 the equation (3) the final pressure p x of the third stage then this 

 pressure and the temperature 273° form the p and T of the equations 

 of the fourth stage. It will frequently happen that the temperature 

 down to which the first stage holds good will lie below the freezing 

 point ; in that case oue passes directly over to the fourth stage, omitting 

 the second and third. After we have thus determined for all the 

 equations the coefficients and the limits for which each equation holds 

 good we can use them in order to determine the T belonging to any 

 given p or inversely. All these computations can however only be 

 executed by successive approximations, and one would do well to take 

 the necessary approximate values from the accompanying diagram. If 

 we have determined p and Tfor any special condition then the remaining 

 characteristics are easily found. The density of the mixture follows from 

 the corresponding equation of elasticity. The equation (la) gives the 

 quantity of water still present in the form of vapor, and therefore also 

 the quantity of water already liquefied. Frequently oue desires to 

 know the difference in altitude h that corresponds to the different con- 

 ditions ^o and p\ under the assumption that the whole atmosphere is 

 found in the so-called condition of adiabatic equilibrium. If one de- 

 sires the exact solution of this problem, it must be attained by the 

 laborious mechanical evaluation of the integral 



h 



= / vdp ; 

 J Pi 



but since it is precisely with regard to this point that an exact deter- 

 mination never has a special value, therefore here one may always 

 abide by the accompanying convenient diagram. 



IT. 



If we had to deal only with one mixture whose composition is exactly 

 known for which we therefore can have only one value of the ratio // : A, 

 then we could exactly re-produce the formula 1 above developed by a 

 graphic table that would enable us to directly perceive the adiabatic 

 changes of the mixture for any condition. 



We should represent pressure and temperature by coordinates in one 

 plane and cover this plane with a system of curves that should con- 

 nect all those conditions together that can adiabatically pass from one 

 to the other. It would then only be necessary to glide from a given 

 initial condition along the curve going through the corresponding point 

 in order to perceive the behavior of the mixture as it passes through all 

 these stages. 



Since however the meteorologist must necessarily deal with mixtures 



