102 Prof. D. Mendeleeff on the Variation in the 



aspect of formula No. 1 bears a certain relation to this 

 subject, as will be seen from the method by which I arrived 

 at it. 



In order to obtain an expression fulfilling the condition that 

 the density of water at 4° be taken as =1, it was necessary in 

 the empirical formula 



S t=a + bt + ct* + df + 



to take a = l and to make the sum of all the remaining 

 members of the series divisible by (£ — 4). But in order to 

 comply with the fact that the density of water attains a 

 maximum at 4°, it was necessary to admit that the remaining 

 members are once more divisible by (£ — 4), because then, 

 when t = 4, the differential coefficient is equal to zero *. There- 

 fore the formula 



S*=l-0-4)*F(0, (2) 



where F(V)>0 and <1, should be taken. Having deter- 

 mined the values of F (t) or the magnitudes (1 — S<)/(£ — 4) 2 

 from the aggregate of existing data, I became convinced from 



* As far as I know (from a notice in Pogg. Ann. 1853, xc. p. 628) 

 Hassler, in America in 1832, was the first to apply an expression of the 

 form 



St=S -A(t-t m )-B{t-t m ) 2 , 



where t m is the temperature of maximum density, for calculating the 

 density of water. The necessity for discarding the term (t—t m )m the 

 first degree, and for taking it only in the second degree, was already 

 recognized by Miller (Phil. Trans. 1856), and is repeated by Hagen and 

 Rosetti. But these observers, in their calculations for the expansion of 

 water according to the formula 



S t =l-(t-t m ) 2 A.+B(t-t m Y+"-C(t-t m )2+*, 



&c, have up till now always only adopted such functions where t 

 invariably has a positive exponent, i. e. only enters into the numerator ; 

 whereas I have become convinced that formulae of this kind satisfy the 

 aggregate of known facts only when taken with a large number of terms, 

 even if fractional indices be adopted, as done by Hagen and Rosetti. 

 And if the number of the terms of the expression' be great, then it loses 

 that simplicity which alone fulfils the requirements we have a right to 

 claim in a natural expression for the phenomena of nature. In addition 

 to this, Hagen expressed the variation of specific gravity by a formula of 

 the aspect 



St = l-(t-t m y[A-B(t-t m )i-^, 



while Rosetti had recourse to a similar formula for the expression of 

 the volumes : 



V t = l+A(t-t m y-B(t-tm) 2 - Q +C(t-t m )f2, 



and since V*S* = 1, a comparison of both expressions, as well as a trial of 

 them, convinces one of the total unsatisfactoriness of both or at least one 

 of them. 



