§ 7 OF ARTS AND SCIENCES. 385 



-" + (2«+l).(| + ±^T), 



so that the value of the quotient must be less than this 

 quantity. Hence the ratio of the (u -|- 2)"^ term to the 

 (;^ -[~ i)^' ^^ the expansion of e cannot exceed the value 



When ;/ becomes indefinitely large, this ratio approach- 

 es the value -— -\- id (— -\- — eZ"), so that if 



— 4- 2d\— A- ■^€.T\<' I, the series is certainly con- 



vero^ent. 



Now t is necessarily less than T^ since 7"° = 273° -f- f\ 

 and the smaller the value of e, the greater can be the ratio of 

 t to T without invalidating the convergence of the series. 



If / < O1 evidently e^ can be as great as one pleases, 

 that is, the series is necessarily convergent for a descending 

 scale of temperature. 



If ^ ^ 60° and e <; .002 



we have 2et l-^ -|- ^ eTj <i ~ <i — - — ; hence 



the tables, which were constructed from e = .0001 to 

 e = .0020, are at least reliable up to 60°. 



In the same way e is certainly determinate up to the 

 value .00135 for at least 90°; and for still smaller values 

 of 60, still higher temperatures may be used. It will be 

 noticed that there is no liquid whose coefficient of expan- 

 sion is as great as .002 which does not boil, at the ordinary 

 pressure, below 60"; and none boiling above 90° has a 

 greater coefficient than .00135, so that thus far theory and 

 fact are not at variance. It must also be remembered 

 that the ratio of a term to the one preceding it was shown 



