Density of Water with the Temperature. 103 



numerous calculations that F (f) is expressed by the sum of 

 the terms a + ht + ct 2 + dt B + . . . . with coefficients consecu- 

 tively changing their signs, i. e. from + to — , and decreasing 

 in magnitude *. This indicated the convergent nature of the 

 series and the possibility of expressing F (t) in a simpler form 

 which would rapidly converge. But I did not consider it as 

 sufficiently exact, for the admission of terms with t 3 and even 

 £ 4 in F (t) did not yet express the entire phenomenon of the 

 expansion of water between — 10° and + 200° with even the 

 small degree of accuracy which is found in contemporary 

 determinations. In striving to express "F (t) in the simplest 

 possible form I tried many of the expressions already proposed, 

 but became convinced of their insufficiency f . As regards 

 formula No. 1, I arrived at it from the following con- 

 siderations : — 



1. When I showed (see note p. 101) that the expansion of all 

 liquids (except water) may be approximately expressed, like 

 the expansion of gases, by the general formula 



kV 



M l+ S)' 



where for gases n=+l and for liquids n=— 1, i.e. when 

 for liquids it was possible to take 



S ( =S„(l-to) (3) 



* As an example I will cite one such formula, performing the multi- 

 plication by (t— 4) 2 in order to show clearly the varying- nature of the 

 signs : 



S*10 6 = 999875+63-606 £-8-3185 £ 2 +O063238 1 3 



-0-00036703 ^+0-0000008979^. 



Expressions with —At e -\-Bt 7 can "be considered as sufficient for the 

 accuracy of contemporary determinations, but the above expression with 

 t% although satisfying the greater portion of the curve, still, for ordinary 

 temperatures (20°-30°) affords deviations which exceed the probable 

 errors in corrected mean values. 



t I for a long time confined my attention particularly to expressions 

 of the form 



a/ 



^ = l + (£-4) 2 F 1 (£), 



where the first member corresponds to the distance between the centres 

 of the particles. I afterwards endeavoured to express the dependence of 

 the density and of the volume of water upon its temperature by taking 

 fractional indices (like Hagen and Kosettij and tried the application of 

 logarithmic (like Rankinej and catenary functions, and in general, like 

 Hagen and Frankenheim, spent much time in endeavours to express this 

 dependence by means of some simple algebraical formula with the least 

 possible number of constants. 



