Hoar-frost Lines of Water-Substance. 513 



The discontinuity and intersection of these lines was pointed 

 out nearly twenty years ago by M. Kirchhoff, in Poggendorff 's 

 Annalen, 1858, ciii. p. 206, in a memoir entitled " Bemer- 

 kung iiber die Spannung des Wasserdampfes Lei Tempera- 

 turen, die dem Eispunkte nahe sind." As this memoir seems 

 to have dropped out of sight (for I have never seen any 

 reference made to it), it may not be without advantage to 

 resume its argument. 



The assumption is made that there is some temperature at 

 which both ice and water can exist, and at which the steam 

 formed over ice is exactly identical • with that formed over 

 water. M. Kirchhoff has taken this temperature as 0° C, 

 though it must be higher by about 0°*0074, as the pressure is 

 only about 4' 6 millims. instead of 760 millims. of mercury. 

 The difference is negligible ; and the temperature will be 

 taken as 0°C, or 27o 0, 7 on the absolute scale, and denoted 

 by t . The argument, then, is, that when unit mass of ice at this 

 temperature is directly transformed into vapour, its intrinsic 

 energy is increased by exactly the same amount as if the 

 transformation were first into the liquid and then into the 

 vapour state. 



Now, during vaporization at temperature t from the solid 

 state, the heat 1/ spent is given by 



where J is the mechanical equivalent of heat, t the absolute 

 temperature, u! the specific volume of ice, and p f y v' the 

 pressure and specific volume of steam over ice. But external 

 work equal to p'(v' —u') is done ; so that the increase of 



intrinsic energy is W —u f )[t~ — p'\. Hence, in the given 



case, the increase is (V — ^oXV^o'-'lV)? the suffixes denoting 



values for t , and ^' being written for -£-. 



Similarly, in the second process, the increase of intrinsic 

 energy during vaporization from the liquid state is 



where u is the specific volume of water at t Qy and p w v the 

 pressure and specific volume of steam over water. Also, 

 during the liquefaction, the increase of intrinsic energy is 

 Jlo+p" (u' — u ), if l is the heat required for the change of 

 state at the pressure p" ; for during the contraction the work 

 Phil. Mag. S. 5. No. 21. Suppl. Vol. 3. 2 L 



