Lags of Thermometers. 65 



the true time-position on the chart. Suppose the tempera- 

 ture inversion in the accompanying figure is under con- 

 sideration, times being measured to the right, and temperature 



towards the top. On the left, the temperature is increasing 

 at the constant rate G, shown by the full line, and the ther- 

 mometer reading lags behind, as shown by the dotted line. 

 After the inversion at P, the air temperature decreases at 

 the constant rate G', shown by the full line on the right. 

 The true maximum is at P, but the thermometer maximum 

 is at Q, where the dotted line crosses the second straight 

 line. In the neighbourhood of P, the lag is given by for- 

 mula (19) — in the case of the cylinder — and we have on 

 writing t=yo in that formula, and retaining only the first 

 term of the series, 



L ^^( 1 + l\ + ^ 2 ^-^y e -a^, (49) 



a 2 V8 + 2H/^/3 1 4 (/3 1 2 + H 2 )a 2e ' { ; 



t' being the time reckoned from the inversion P. The 

 value, 8, of t', at which the dotted line crosses the full line 

 in the figure, gives the displacement of the temperature 

 maximum and for this value the lag L is zero. Hence to 

 determine 8 we have the equation 



4H ! (G-^) 5 



a* id 



Now for cases in which the first term of the series gives 

 the lag with sufficient accuracy the value of 



is very nearly equal to unity. For example, we have the 

 Phil. Mag. S. 6. Vol. 43. No. 253. Jan. 1922. F 



