496 Dr. J. E. Mills on the 



expressing a certain relationship which they actually do 

 possess. 



Again, it is possible to approximate any simple curve more 

 or less closely by a number of quite different equations. 

 Thus the vapour-pressure curve has been approximately 

 reproduced by various equations. But these equations for 

 the most part make no attempt or pretence to represent 

 nature. So far from all of them being true they are none 

 of them exactly true. If one could be found exactly true, 

 then its variables and constants would have a more than 

 mathematical significance ; that is, they would represent 

 certain variables and constants of nature which we might 

 say " determine " the vapour-pressure curve. If Kleeman 

 means that as a mathematical exercise he can produce by 

 the use of arbitrary constants various equations connecting 

 the densities and internal heats of vaporization with an 

 approximation to the truth. I will grant it. But if he means 

 that in my work I have overlooked a constant of integration 

 I think he is wrong. 



Considering equation (5) as representing per se the relation 

 between the internal heat of vaporization and the density 

 of the substance, it is clear that from equation (5) alone it is 

 impossible to deduce the fact that an infinite number of 

 different relations exist between p and \. Because such a 

 deduction would place us in the position of being able to prove 

 that because one relation existed between tioo quantities, an 

 infinite number of relations must exist between the same tivo 

 quantities. 



6. It remains to locate the exact errors in Kleeman's 

 proof of his position. Without quoting his proof, and 

 proceeding on the supposition that it is available to those 

 interested in locating the error, I deal first with the proof 

 when the internal heat of vaporization L is expressed as a 

 function of the density of the liquid and the temperature 

 according to the equation h = yjr 2 (p, T). (Kleeman's symbols 

 are used in this section.) Kleeman's first step is to draw a 

 curve A X A 2 (fig. 1) to represent this equation at the 

 temperature T x . Now as a matter of fact for a particular 

 substance, at a particular temperature, there is but one 

 value of the heat of vaporization and Kleeman's curve should 

 be a point. 



Considering next the proof when h = ty 2 (p), and A 1 'A 2 / , 

 fig. 2, its graph. Kleeman then says : " Let the difference 

 between the ordinates c^, 6 1? b 2 denote the latent heat of a 



liquid corresponding to the temperatures T l9 T 2 etc/' 



Since there is in reality a relation between p and T (a relation 



