Relation to Pressun and Temperature. 499 



When p is small, y=d-p(l — ap/2), which will be further 

 simplified in the next paragraph. 



The expression for compressibility has a counterpart* which 

 is applicable for thermal expansion, viz : 



This application must be omitted here. 



Finally the change of p when regarded as a function of the 

 radius p of any unit sphere within the compressed liquid is of 



interest. It appears that p=(? ~~l)/«. This equation 



points out the nature of the inadequacy of (2) ; for if p de- 

 creases indefinitely, p eventually becomes v ea/v —l)/a. In the 

 next paragraph a/d- is found to be nearly 9. Hence the limit 

 in question is 8290/«. In the cases where (2) applies this 

 value lies somewhere between 10 6 and 10 7 atmospheres. Hence 

 the interval within which (2) may apply satisfactorily is rea- 

 sonably large ; for the pressures stated are such as would be 

 met with at the center of the earth for instance. 



29. Direct computation. — After the suggestions contained 

 in § 27, approximate values for the constants in the equation 



a / 



v/ V=ln(l + ap) 7 are easily derived. f Constants thus obtained 

 are crude. Hence starting with these, I computed more accu- 

 rate data by a method of gradual approximation, finally selecting 

 such values of # and a as reduced the errors to a reasonably 

 small amount. This computation is exceedingly tedious and 

 unsatisfactory at best, because pairs of values of # and a 

 differing very widely from each other, are often found to sat- 

 isfy the equation about equally well. Nevertheless it was 

 necessary to avoid any scheme of selection other than the 

 criterion of errors specified, the object being to obtain a set of 

 exponential constants independent of ulterior purposes or con- 

 siderations. The results are given in table 21, in which the 

 first column contains the initial pressure p for which # and a 

 apply at the temperature 0. The table also contains the ratio 

 23 #/a, the factor reducing it to common logarithms. At the 

 end of the table I have added the mean datum 2 3 # /a , 

 derived from all the values of the table. 



30. Investigation of mean constants. — The tabulated con- 

 stants for ether and alcohol above 185°, substantiate the 

 remarks made in § 28, relative to the limited high temperature 

 application of v / V=ln (l + ap)^/ a . The critical temperatures 



* Applying this equation to the isopiestics of ether between 30° and 310°, I 

 found the observations well reproduced, in a way to include the remarkable 

 volume changes at and above the critical temperature. 



f Reference to y is conveniently dropped here. 



