Vapour-pressure and the Volumes of Vapour and Liquid. 389 



To calculate the table, I first represented X by a series pro- 

 gressing by powers of a quantity dependent on 9. For this 

 the following quantity appeared to me the most suitable, 



x = s/T^~0; (25) 



which, like X, when the critical temperature is approached, 

 approaches the value nil. The series in question is 



\=-6«+ 3-24^ 3 + 2-8801716^ 5 + 2-885628^ 7 + . . . . (26) 



Before speaking of the application of this series to the cal- 

 culation, a consequence resulting even from its form may be 

 interpolated, which is connected with the remark made at the 

 close of the preceding section. As is seen, the series contains 

 only odd powers of x; and thence it immediately follows that 

 the series which represents X 2 can only contain even powers 

 of x. Since, further, as mentioned above, the quantity II, 

 when the expansion takes place with respect to X, contains 

 only even powers of X, it can only contain even powers of x, 

 according to the foregoing, when expanded with respect to x, 

 while the series representing the quantities w and W contain 

 also terms with odd powers, and among them a term of the 

 first order. We now get from (25), for the differential coeffi- 

 cients of x and x 2 with respect to 6, the following expressions, 



dx, 





cw 2Vi-e' v . . . (27) 



d9 ~ ' 



which as the critical temperature is approached (for which 

 = 1) differ essentially from each other in their behaviour, 

 in that the former becomes infinitely great, while the latter 

 remains finite. In just the same way must, according to what 

 has been said above, the differential coefficients of the quanti- 

 ties w andW, taken with respect to 6, differ from the differen- 

 tial coefficients of the quantity II. It may here be likewise 

 added that the same holds good also for the differential coeffi- 

 cients taken with respect to T; and from this it follows that on 

 approaching the critical temperature the volumes of the liquid 

 and the vapour undergo changes which are infinitely great 

 in proportion to the change of temperature. Van der Waals 

 has already called attention to this singular difference. 



With the help of the above series I have calculated X for 

 those values of 8 and x for which that number of terms suffices 

 for attaining the desired degree of accuracy. For higher 

 values of x, and consequently lower values of 0, I returned to 



