434 PROCEEDINGS OF THE AMERICAN ACADEMY. 



tial dissimilarity between a liquid and a crystal on the other hand. 

 The molecules in both the liquid and the vapor are thought of as 

 arranged at random, " molekular-ungeordnet," the only difference be- 

 tween liquid and vapor lying in the average distance apart of the mole- 

 cules. Once let the average distance apart of the molecules appropriate 

 to the vapor and licjuid become equal, as they do at the critical point, 

 and all further distinction between the two phases vanishes. In par- 

 ticular, the heat of transformation vanishes as a consequence of the 

 equality of the volumes. But for the two phases liquid-crystal, there 

 is besides the distinction of density the further distinction that in the 

 crystal the molecules are arranged in some sort of a framework in 

 space. Equality of density of the two phases liquid and solid does not 

 mean identity, because the crystal still preserves the regular arrange- 

 ment of the molecules w'hich cannot hold in the liquid. In conse- 

 quence, the energy difference between the two phases need not become 

 zero when the volume difference vanishes ; that is, the heat of trans- 

 formation does not vanish at the same time with the change of volume, 

 and we do not have a critical point. Tammann goes further than 

 simply to deny the existence of a critical point. He concludes from 

 an examination of all the known data, chiefly his own experiments, that 

 for substances of the ordinary type of freezing with decrease of volume, 

 not only do the change of latent heat and the change of volume fail to 

 pass through the value zero at the same time, but that the change of 

 volume always becomes zero before the heat of transformation. It is 

 then an immediate deduction from Clapeyron's equation that the melt- 

 ing curve has a maximum. That is, there exists for every substance a 

 temperature so high that no pressure, however intense, will bring the 

 point of transition as high as this temperature. Above this tempera- 

 ture the substance is always liquid. This is really the essential part of 

 Tammann's theory of the shape of the equilibrium curve. It has been 

 immediately seized upon by geologists, and made the basis of many 

 speculations as to the state of the interior of the earth, the matter 

 there being supposed fluid because the temperature is higher than 

 the maximum. 



The experimental evidence in support of this theory is meagre, the 

 pressure reached by Tammann being sufficient only to show a cur- 

 vature in the direction demanded by the presence of a maximum, 

 and an initial decrease toward zero of the change of volume more 

 rapid than the decrease of the latent heat. The only case in which 

 Tammann claims to have found a maximum, for Na2SO4l0H2O, must 

 be ruled out because of the complication introduced in the equilibrium 

 conditions by the water of crystallization. 



