386 PROCEEDINGS OF THE AMERICAN ACADEMY § 7 



t 1 A. 



to be less than -— - -f" 2e/ (— -|- — eT") by assuming that 



the maximum ratio of the corresponding coefficients held 

 for all. Since this is evidently not the case, the series is 

 much more convergent than has been assumed, and there 

 is every reason to believe that the value of e is deter- 

 minate for all temperatures below a certain critical point. 

 Such a point is now believed to exist for all substances, 

 above which the laws of liquid expansion will cease to 

 apply. The correspondence of theory and fact in respect 

 to this point will, if established, afford a complete physi- 

 cal demonstration of the sufficient convergence of our 

 series. 



The value of the critical temperature we are not yet 

 prepared to calculate, owing to the difficulty of the mathe- 

 matical solution; but in one of the sections following we 

 shall see how it may be derived from our theory in a 

 much more simple way. 



It remains, therefore, to conclude that, since e is deter- 

 minate for all points below the critical temperature, the 

 volume and its coefficients must also be determinate, and 

 their calculation by means of the derivatives in Table I. is 

 therefore perfectly legitimate. 



Equations VII. enable us to determine the value of ^, b^ 

 c, d and e, successively, for all values of e at the temper- 

 ature zero; and a table will be found at the end of the 

 paper (II.) in which these coefficients are calculated for 

 values of 60 from .0001 to .0020. 



By means of these coefficients, a volume table has been 

 constructed (III.) by which, when the value of e^ is known, 

 the volume may be found, calculated at intervals of 10°, from 

 — 10'' to 150" centigrade, the volume at zero being unity; 

 and furthermore, by means of this table, if the volume be 

 known at any temperature besides zero, the value oi e^ and 

 hence the volume at any other temperature may be de- 

 duced. 



