380 PROCEEDINGS OF THE AMERICAN ACADEMY § 7 



a relation between e and its lirst derivative which is inde- 

 pendent of the value of / or /'. 

 DifFerentiatinc asrain, we find 



dT' 3 dT ^ ^ dT ^ I 



€^ 



where, substituting the value of ~^ in terms of e, we 

 have 



-^ = 12.263 + 15.5^^^+ S-Z^'T\ HI. 



In the same way, by successive differentiation and sub- 

 stitution of the values of the first derivative in terms of e, 

 we have 



d^e 



-- loi.iie'^ -f- 2o\r^^feT A^ 145.196^7'' 



dT^ 



+ 35-55^^ n IV. 



accurate to two places of decimals, and 



-^= 1148.6^ + 3218.6^ r+ 3504.^^^^ + 1743.^'^^ 



+ 332.6^r^ V. 



where the coefficients are expressed to the nearest unit; 

 so that when e is given, the values of its successive deriva- 

 tives at any temperature arc determined. 



At the end of this paper will be found a table (I.) show- 

 ing the corresponding values of e and its first four deriva- 

 tives at the freezing temperature of water (7^0=273"), 

 from 60= .0001 to e„^ .002. Tables may easil}^ be con- 

 structed for any temperature; the use of this particular 

 table is explained below. 



Let the volume ( F) of a liquid which is unity when the 

 temperature is 0° centigrade (/ = o) be represented by a 

 sufficient number of coefficients in the form, 



V— I + t?/ + bt' -|- cl' + dt' + en + etc.; 



