Barrett — The Expansion and the Densities of Liquids. 329 



The long cylindrical thermometer bulb saves the necessity for 

 frequent stirring of the warm water, as any differences of tem- 

 perature in the water are summed up by the thermometer bulb 

 extending through the different strata ; hence the reading of the 

 thermometer gives the true mean temperature of the whole mass of 

 water. After thermal equilibrium has been established between 

 the liquid in the U tubes and the water around them, the vertical 

 difference of level, CD and CD', in each tube is carefully read by 

 means of a kathetometer, and the thermometer in each vessel also 

 read. Another pair of temperatures and another pair of differences 

 of level are quickly obtained, and furnish a second experiment with 

 the same liquid. To enable the liquid to be easily changed, the 

 U tubes are fastened to a cross piece of wood, W, which is fixed to 

 the upright, T, by buttons s, z : turning the buttons allows the U 

 tubes to be removed from the vessels A, B, when the liquid can be 

 emptied and the tubes cleaned. 



The calculation of the coefficient of expansion is very simple. 

 The pressure on the surface of the liquid at C and C, being that 

 of the air common to the two, is the same ; the pressure on the 

 surface of the liquid at D and D', being that of the atmosphere, is 

 also the same, and greater than that at C, C by a column of liquid 

 equal to the vertical distance between C and D, or C and D' . 

 Hence, as the whole is in equilibrium, the pressure of the liquid 

 column, CD, is equal to the pressure of CD' ; but the liquid in B 

 is warmed, and its diminished density, consequent on its dilatation, 

 will cause the column CD' to be longer than CD by an amount 

 proportional to its dilatation ; this, again, is directly proportional to 

 the coefficient of absolute expansion of the liquid to the length of 

 the column CD', and to the difference of temperature between A 

 and B. Let k be the coefficient of absolute expansion of the liquid, 

 h the difference of level CD, and h' the difference CD' ; t being the 

 temperature of the water in A, and t' the temperature in B. Then 



h' -h = k. h' [f - t), 

 whence, 



_ h' - h 



h'{t'-t)' 



The atmospheric pressure is assumed to be constant throughout the 

 course of an experiment, but an error is apt to arise from the hot 



