82 Mr. H. C. Sorby on the Expamion of Water and 



observed. In the same volume of the work just cited, Franken- 

 heim shows that the expansion of many hquids may be expressed 

 with sufficient accuracy by means of equations of the form 



V = A + B^ + a*, 



but says that here and there the regular progress of the differ- 

 ences between the results calculated from the formulae and the 

 observations shows that another term, involving ^, might be 

 employed wvi\\ advantage. My own experiments lead me to 

 agree to a great extent with this last remark. They show that 

 probably the expansion really follows a law expressed by a 

 series involving ascending powers of t, but the coefficients of 

 higher powers than the second have such small values, that, for 

 a very considerable range of temperature, they have scarcely any 

 more influence than the unavoidable errors of observation. 



In discussing the relative volumes at various temperatures, 

 the method of successive differences can be made use of with 

 very great advantage, if the volume be known at equal intervals 

 of temperature. So far, however, as I have been able to learn, 

 previous observers have not employed it, either from not having 

 thought of it, or from having considered it preferable to ascer- 

 tain the volume of the liquid at various accidental temperatures, 

 without endeavouring to raise the heat to particular given tem- 

 peratures. If the law of the expansion of a liquid be really of 

 the form 



V=A + B<-|-C/2 + D/3 + &c., 



and if the volume at a number of equidistant temperatures be 

 known, it is easy to see what terms of the series should be taken 

 into consideration, by determining the successive differences of 

 the volumes. From the well-known properties of such a series, 

 if each volume subtracted from that next above it give these 

 first differences of the same value throughout, there can be no 

 term involving a higher power of t than the first. If, however, 

 these first differences increase in magnitude from a low to a 

 higher temperature, but, when we subtract each of them from 

 that next above it, if we find that these second differences are of 

 a constant value, we may be equally sure that there is a term 

 involving /-, but no higher powers. The same law holds good 

 for ^, i^, &c. ; and therefore, as will be seen, this method is so 

 exti'emely suitable for the inquiry now before us, that in my ex- 

 periments I ascertained the volume of the liquids at given equi- 

 distant temperatures, in order to be able to make use of it. 



Before commencing my own experiments, I discussed by this 

 method of successive differences the relative volumes of water at 

 various equidistant temperatures, deduced by Kopp from the 



