Steam of Maximum Density. 171 



which formula, when expressed in series, becomes 



logP,= A^l-M/ + M 2 * 2 -M 3 * 3 +&c.}. 



The earliest of the empirical formulas referred to is that of our 

 own countryman Dalton, who suggested, as the foundation of 

 useful formulae of interpolation, the principle, that the elastic 

 forces increase in geometrical progression, whilst the tempera- 

 tures increase in arithmetical progression. Thus Dalton's for- 

 mula of interpolation would be 



logP,=A*, 



a formula consisting of one term only, and that term being the 

 first of the series (above given) constituting the Roche formula. 

 M. Regnault (at page 587 of the volume* cited) makes a 

 valuable remark which is applicable to the two empirical formulas 

 just mentioned, as well as to other empirical formulas relating to 

 steam. The remark is, that such formulas express correctly only 

 the first, or only the first and second terms of the series consti- 

 tuting the true formulas. That is to say (applying and extending 

 the principle), if the true formula, expressed in series of ascend- 

 ing powers of t 3 were 



log P,= A* + B* 2 + Ct 3 + &c, 



then the approximate formula of the first degree would be 

 \og¥ t = At, which is that of Dalton. And the approximate 

 formula of the second degree would be log V t = At 4- Btf 2 + &c, of 

 which Roche's formula is an example, the first and second co- 

 efficients A and B being identical with the first and second 

 coefficients of the formula expressing the true law. Proceeding 

 in the same course, it may safely be said that if an approximate 

 formula of the third degree were discovered, it is highly pro- 

 bable that such formula would coincide with the true formula, 

 not only in the first three terms, but in all the succeeding terms 

 of the two series, and that such formula would be as great an 

 improvement on the Roche formula as the latter is an improve- 

 ment on the formula of Dalton. 



The general character of the law of variation according to tem- 

 perature of the elastic force of steam of maximum density, may 

 be easily perceived on inspection of the tabulated values of these 

 forces as given either by MM. Aragoand Dulong or by M. Reg- 

 nault. If the logarithms of these forces, separated by equal 

 intervals of temperature (as 5 or 10 degrees Centigrade), be 

 written under one another in a vertical column, and the differ- 

 ences of the consecutive logarithms be taken, the series thus 

 formed of the quantities A log P will evidently be composed of 

 terms uniformly decreasing with the temperature, from the very 



* Memoires de VAcademie, &c, vol. xxi. 



N2 



