§ 8 OF ARTS AND SCIENCES. 387 



If, moreover, only the ratio of expansion be known 

 for a given range of temperature, it w^ill be possible to find 

 b}^ trial what value of e^ will account for this ratio, and 

 thus to determine completely the laws of expansion of 

 the substance. 



The advantage of this method of treating expansion, 

 which i-equires for substances not subject to a change of 

 state onl}' a single accurate observation in place of at least 

 three, and which is the same for all bodies having the fun- 

 damental coefficient in common, is obvious; it remains 

 onl}' to see how closely our calculations are borne out by 

 experience. 



§ 8. The usual method of determining the law of ex- 

 pansion of a substance depends upon the assumption that 

 the volume at any temperature, /, can be expressed suffi- 

 ciently well by three constant coefficients. A, jB, C, in the 

 form 



Four observations of the volume are usually made at dif- 

 ferent temperatures, the first being preferably zero, or, when 

 that is impossible, the melting point. The other three, in 

 absence of the data, may be assumed to have been chosen 

 at equal intervals up to the boiling point, or the highest 

 temperature observed. It is then always possible to 

 assign such values to A^ B and C as shall make the vol- 

 umes calculated by the formula agree with those observed 

 at all four temperatures. The values of the coefficients, A^ 

 B and C, as determined by Kopp and Pierre, have been 

 tabulated by Sharpies for about eighty-eight liquids, of 

 which eleven were determined by both observers. In 

 order to compare our results most directly with their fig- 

 ures, we must calculate the theoretical values of yl, B and 

 C which would represent the volumes correctly according 



