xxi.] THEORY OF APPROXIMATION. 471 



tinuity to the conception of a perfect gas. Such a gas 

 would probably consist of atoms of matter at so great a 

 distance from each other as to exert no attractive forces 

 upon each other ; but for this condition. to be fulfilled the 

 distances must be infinite, so that an absolutely perfect 

 gas cannot exist. But the perfect gas is not merely a 

 limit to which we may approach, it is a limit passed by 

 at least one real gas. It has been shown by Despretz, 

 Pouillet, Dulong, Arago, and finally Regnault, that all 

 gases diverge from the Boylean law, and in nearly all 

 cases the density of the gas increases in a somewhat greater 

 ratio than the pressure, indicating a tendency on the 

 part of the molecules to approximate of their own accord. 

 In the more condensable gases such as sulphurous acid, 

 ammonia, and cyanogen, this tendency is strongly apparent 

 near the liquefying point. Hydrogen, on the contrary, 

 diverges from the law of a perfect gas in the opposite 

 direction, that is, the density increases less than in the 

 ratio of the pressure. 1 This is a singular exception, the 

 bearing of which I am unable to comprehend. 



All gases diverge again from the law of uniform ex- 

 pansion by heat, but the divergence is less as the gas in 

 question is less condensable, or examined at a temperature 

 more removed from its liquefying point. Tims the perfect 

 gas must have an infinitely high temperature. According 

 to Dalton's law each gas in a mixture retains its own 

 properties unaffected by the presence of any other gas. 2 

 This law is probably true only by approximation, but it 

 is obvious that it would be true of the perfect gas with 

 infinitely distant particles. 3 



Mathematical Principles of Approximation. 



The approximate character of physical science will be 

 rendered more plain if we consider it from a mathematical 

 point of view. Throughout quantitative investigations we 

 deal with the relation of one quantity to other quantities, 



1 Jamin, Cours de Physique, vol. i. pp. 283 288. 



2 Joule and Thomson, Philosophical Transactions, 1854, vol. cxliv. 



P- 337- 



3 The properties of a perfect gas have been described by Rankine, 

 Transactors of the Royal Society of Edinburgh, vol. xxv. p. 561. 



