SECT. II.] ERROR OF A THERMOMETER. 283 



or u Ae~ m and -j- = hv + hAe~ Ht 



at 



lead to the equation 



v u = le~ ht + aHe~ m , 



a and Z&amp;gt; being arbitrary constants. Suppose now the initial value 

 of v u to be A, that is, that the height of the thermometer 

 exceeds by A the true temperature of the fluid at the beginning 

 of the immersion; and that the initial value of u is E. We can 

 determine a and b, and we shall have 



The quantity v u is the error of the thermometer, that is 

 to say, the difference which is found between the temperature 

 indicated by the thermometer and the real temperature of the 

 fluid at the same instant. This difference is variable, and the 

 preceding equation informs us according to what law it tends 

 to decrease. We see by the expression for the difference vu 

 that two of its terms containing e~ u diminish very rapidly, with 

 the velocity which would be observed in the thermometer if it 

 were dipped into fluid at constant temperature. With respect 

 to the term which contains e~ Ht , its decrease is much slower, 

 and is effected with the velocity of cooling of the vessel in air. 

 It follows from this, that after a time of no great length the 

 error of the thermometer is represented by the single term 

 HE H 



e -Ht or 



h-H h-H 



299. Consider now what experiment teaches as to the values 

 of H and h. Into water at 8 5 (octogesimal scale) we dipped 

 a thermometer which had first been heated, and it descended 

 in the water from 40 to 20 degrees in six seconds. This ex 

 periment was repeated carefully several times. From this we 

 find that the value of e~ h is Q 000042 1 ; if the time is reckoned 

 in minutes, that is to say, if the height of the thermometer be 

 E at the beginning of a minute, it will be #(0-000042) at the 

 end of the minute. Thus we find 



ftlog l0 e = 4-376127l. 



1 0-00004206, strictly. [A. F.] 



