2(36 PROFESSOR TAIT ON THE 



portion which is practically straight and parallel to the axis of volume. That this 

 conclusion is apparently borne out by experimental facts (so far at least as these are not 

 modified by the residual trace of air) will be seen when we make the comparison. 



In fact we might speak of a superior and an inferior critical volume, and the portions 

 of the isothermals beyond these limits on both sides may perhaps have equations of the 

 same form, but with finite changes in some at least of the constants. 



Another source of a species of discontinuity in some, at least, of the constants is a 

 reduction of E to such an extent that grouping of the spheres into doublets, triplets, &c, 

 becomes possible. Thus we have a hint of the existence of a " critical temperature." 



It must be confessed that, while we have only an approximate knowledge of the length 

 of the mean free path (even among equal non-attracting spheres) when it amounts only to 

 some two or three diameters, we practically know almost nothing about its exact value 

 when the volume is so much reduced that no particle has a path longer than one diameter. 



[It might be objected to the equation arrived at above, should it be found on com- 

 parison with experiment that a and y are both positive, that it will not make p infinite 

 unless v vanish. To this I need only reply that the equation has been framed on 

 the supposition that the particles are in motion, and therefore free to move. What 

 may happen when they become jammed together is not a matter of much physical 

 interest, except perhaps from the point of view of dilatancy. If the equation 

 represents, with tolerable accuracy, all the cases which can be submitted to experi- 

 ment, it will fully satisfy all lawful curiosity.] 



XXL' — Relation between Kinetic Energy and Temperature. 



70. Before we can put the above virial equation into the usual form of a relation 

 among p, v, and t, it is necessary that we should consider how the temperature of 

 an assemblage of particles depends upon their average kinetic energy. 



Van der Waals and Clausius, following the usual custom, take the average kinetic 

 energy as being proportional to the absolute temperature. Clerk-Maxwell is more 

 guarded, but he says : — " The assumption that the kinetic energy is determined by 

 the absolute temperature is true for perfect gases, and we have no evidence that 

 any other law holds for gases, even near their liquefying point." 



On this question I differ completely from these great authorities, and may err 

 absolutely. Yet I have many grave reasons on my side, one of which is immediately 

 connected with the special question on hand. To take this reason first, although it 

 is by no means the strongest, it appears to me that only if E above (with a constant 

 added, when required, as will presently be shown) is regarded as proportional to 

 the absolute temperature, can the above equation be in any sense accurately con- 

 sidered as that of an Isothermal. If the whole kinetic energy of the particles is 

 treated as proportional to the absolute temperature, the various stages of the gas 

 as its volume changes with E constant correspond to changes of temperature with- 



