240 DE. A. MATTHIESSEN OX THE EXPANSION 



of the piece of glass rod in vacuo and in water at the different temperatures. In the 

 third the numbers represent those proportional to the volumes of waters displaced at 

 the different temperatures. These values were arrived at in the following manner : — 



Let a be the coefficient of the cubical expansion of the glass rod*, and V„ its volume 

 at 0°. Its volume at t° will of course be Yg(l-\- at). But this is likewise the volume 

 of the displaced water whose Avcight, say W^, is determmed by experiment. Hence 



Vo(l+aO 



is the volume, at f, of the unit of weight of water ; or more generally the vobnne of a 

 given tveight of water is i^rojyortional to 



\+at 



In the fourth column the volume of water is taken at 4 equal to 1° (Table IV.) ; and 

 to reduce the values in Tables V. and VI. to the same unit, a formula was calculated 

 from the first five observations in Series I. Table IV. to express the expansion of water 

 between 4° and 20"-4. 



The formula from which the calculated values (Table IV.) were obtained was 



V,= l-fO-0000084465(i-4)^-0-00000008900(#-4)\ 



and calculating from this formula the volumes at 5°"35 and 8°"90, and takmg them equal 

 to the values found for the numbers proportional to the volumes of water at the same 

 temperatures in Series II. and III. (Tables V. and VI.), we reduce these volumes of water 

 at different temperatures to the same unit. 



Table VII. contains the values thus obtained for the expansion of water from the 

 three series. 



* «= 0-00002187. The coefficient here used must naturally be that obtained for the glass rods after having 

 been heated, for the pieces were boiled with the water before the weighings were made. 



