Density of Water with the Temperature. 105 



where a and b are positive and c negative; for then ABC = a, 

 (B-A)C = 5, and -C = c, for 



{A + t)(B-t)G=ABG + (B-A)Ct-Ct\ 



Besides, under the above conditions l/<j>{t) or ¥(t) should 

 expand in terms of t into a convergent series with changing 

 signs, as is obtained in reality. 



By taking from the most trustworthy determinations the 

 values of <j)(t) corrected as far as possible for temperatures 20°, 

 30°, 40°, 50°, 60°, 70°, and 80°, and employing the method of 

 least squares *, I obtained: — 



0(0 = 125780 + 1158 *-l-90 1 2 , ... (5) 



and the mean quadratic error of calculation proved to be far 

 less than the possible error of experimental results. As a 

 final verification of the formula obtained, the values of S* were 

 extrapolated by means of the expression No. 5 throughout the 

 range of temperatures from —10° to 200°, and it was found that 

 the difference between the values obtained by experiment and 

 calculation in no case exceeded the errors which must be re- 

 cognized as existing in the determinations of the density of 

 water. The figures thus obtained are given in Table III. 



5. As the extrapolation was extended beyond the range of 

 my calculations (from 20° to 80°, = 60°) up to limiting tem- 

 peratures exceeding more than three times the one adopted 

 (from -10° to 200°, = 210°), and as formula No. 1 justified 

 itself by a possible concordance with experimental results, and 

 since the accuracy of existent determinations is very dissimilar 

 and generally speaking small, I considered it useless, pending 

 the publication of more accurate determinations, to search for 

 a more trustworthy value of (j)(t) or through it of the value of 

 S , taking the aggregate of all contemporary data; and this all 

 the more, seeing that for ordinary temperatures (from 0° to 

 40°) the values tj>(t) and S found from formula No. 5 were 

 entirely satisfactory. Taking into consideration the fact that 

 in the expression 



y = a + bt + ct* 



the values of the parabolic coefficients, a, b and c, deduced from 

 experimental data by the method of least squares, are greatly 



* In all my calculations, when it was necessary to adopt the method 

 of least squares I used the process of computation based upon P. L. Tcheby- 

 shetf's method, which is fully explained in my work upon " The Com- 

 pounds of Alcohol and Water," 1865, p. 89. 



