36 SCIENCE PROGRESS 



others to obtain a closer agreement with the experimental 

 results either by the inclusion of an additional constant or 

 better by making them functions of the temperature and per- 

 haps of pressure. Probably the most satisfactory of these is 

 that due to Reinganum 



(5) ('+5)^-" 



where a 1 and b l are functions of both v and T. It is certainly very 

 exact for comparatively small densities, gives a good agreement 

 for higher densities, and is capable of easy manipulation. 



The other main line of development is more empirical, al- 

 though many points have to be considered before the best form 

 is reached. It is clear that the corrections to (2) which are 

 given by (4) or (5) could be covered by a convergent series in 

 powers of the density in which the coefficients of the various 

 terms were determined from experimental data. There is much 

 to be said for expressing the product pv as a series of increas- 

 ing powers of d or -. The series developed by H. K. Onnes 



principally from the experimental results of Amagat is 



(6) . . . . pv = A + B/v + C/v* + D/v 4 + Ejv* + etc. 



in which p and v are most conveniently expressed in mega- 

 dynes and theoretical normal volumes, at constant temperature. 

 It is found that with the highest pressures used by Amagat 

 (about 3,000 At) when the density is about io 3 the F term is the 

 last that is necessary. 



For every substance it is clearly possible to obtain such 

 a series with some accuracy, if the measurements cover a 

 sufficiently wide range, thus enabling the relations between 

 p and v to be known at certain given temperatures. To obtain 

 the change with temperatures a number of isothermals at differ- 

 ent temperatures are required, the change of coefficient between 

 any two being sufficient to give the relation over that particular 

 range. 



However, it is the combination of these relations with the 

 principle of corresponding states which makes their use parti- 

 cularly instructive. The equation (4) can be put into the 

 reduced form in which the pressure volume and temperature 

 are generalised and a and b vanish by noting that, as it is a 

 cubic equation in v, it will have three roots, which must all 



