§ 20 OF ARTS AND SCIENCES. 429 



course very much smaller than the probable error of ob- 

 servation. 



Table XVII. has been constructed to show the probable 

 error when all the liquids quoted are taken into account, 

 and to compare this error with that of the Theory. It will 

 be seen that the average error is here in the tens and not in 

 the units, and that the probable error of the Theory is a 

 little smaller, and could be made very much so, had it 

 been thought advisable to introduce the empirical constant, 

 o. 1 16, instead of that calculated from purely theoretical 

 considerations (.11876). 



The use of either formula is of practical importance, and 

 the difference between their indications is no greater than 

 the limit of error of observation. This is owing; to the 

 close agreement, at low temperatures, of the formulae for 

 expansion (already considered), by which, in combination 

 with the principle of Van der Waals, the critical tem- 

 perature might be calculated in each case. 



Both formulae agree in indicating, for all liquids, an in- 

 variable ratio between the densities at the absolute zero 

 and at the critical temperature. From the empirical 

 formula (II.), we see that this ratio should be <7 -^ 

 {a — i), or 2.005, nearl}^ while the Theory requires 

 the cube of /, -^ /,' or about 3.4. Various considera- 

 tions, whose discussion would be unprofitable, tend to 

 diminish this theoretical value; but a single accurate 

 determination of the density of a liquid in the critical state 

 would probably serve as a crucial test of the reasoning. 



§ 20. Conclusion: — (i) It is not claimed that the 

 theory which has been developed is by any means a com- 

 plete solution of the relations existing between the various 

 constants of Thermodynamics; the main point has been to 

 show that such relations exist. The provisional assump- 

 tion of a cohesive force varying inversely as the fourth 



