HARPER: THERMO METRIC LAG 203 



temperature and the time when the thermometer indicates this 

 temperature; i.e., the number of seconds the thermometer lags. 



2. If a thermometer be plunged into a bath maintained at a 

 constant temperature (the thermometer being initially at a dif- 

 ferent temperature), X is the number of seconds in which the dif- 

 ference between the thermometer reading and bath temperature 

 is reduced to e _1 times its initial value. 



From a series of thermometer readings (6) obtained at suffi- 

 ciently close intervals, the variation of 6 with t may be considered 

 as given and the value of the temperature (ui) corresponding to 

 any thermometer reading (0i) is at once obtainable by employ- 

 ing (1) in the form 



dd 



tti = 0i + X 



at Ji 



(2) 



Conversely 6 may be obtained from u, i.e., the behavior of the 

 thermometer in a bath whose temperature varies with time accord- 

 ing to some given law, can be predicted from the solution of the 

 differential equation. The general solution is 



i i r» ( 



*J o 



d-U=(d -U )e ^ -e *' | —e^dt (3) 



and can be much simplified for the most important special cases 



du 

 by employing the appropriate value of — . For instance if the 



bath be maintained at constant temperature 



i 

 (d - u ) = (e - u ) e x ' (4) 



and if the temperature of the bath rise linearly with time (at rate 

 r) the relation 



u — 6 = r\ (5) 



will hold a short time after the introduction of the thermometer 

 into the bath. 



These equations are the source of the interpretations of X given 

 above. Other cases may be likewise developed, the results being- 

 somewhat more complex than (4) or (5). The equations furnish 



