430 WHITE: PRECISION OF THERMOSTATS 



simplified, is as follows: 



A^P = A0B + VL, 

 where Adp is the periodic oscillation performed by the bath as 

 the heat goes off and on ; L is the lag ; V is the rate of tempera- 

 ture change due to the interrupted, or regulated, part of the 

 heating. A^p is ordinarily the largest variation or error in the 

 bath temperature. This equation, therefore, measures the effi- 

 ciency of the thermostat. It reveals one fact which is true even 

 where the equation does not apply, namely, that the rate of 

 heating, V, is quite as important as the lag, L. 



My other paper presented a formula, more rigorous but more 

 complicated that Sligh's, which often gives results very different 

 from his. 



It was further pointed out, however, that where the two for- 

 mulas disagree seriously, neither one applies, as a rule. The 

 reason is that both treat the lag as a single quantity, which is 

 equivalent to assuming that the fluid in the bulb is always at a 

 practically uniform temperature. But for rapid alternations 

 of bath temperature the temperature change travels into the 

 bulb as a damped wave, and the middle portions of the fluid 

 may take no appreciable part whatever in the periodic oscilla- 

 tion. It is evidently proper to use a formula based on the damped 

 wave; a sufficient approximation, probably, is the well-known 

 formula for plane waves due to harmonic oscillations: 



e = doe-"' sin f -? t - ax) (i) 



where 20o is the temperature range at the margin ; T the period ; 



~ 7. O'T) 



X distance measured into the body; a = Xy-i^, with h^ the 



diffusivity. A very probable value of V is o.oi° per minute; 

 with it a temperature fluctuation of o.ooi° may correspond to 

 an oscillation of 24 seconds period. The effective penetration 

 of such an oscillation into gasoline or toluol is of the order of 

 0.5 mm., and the dimensions of the bulb must be chosen ac- 

 cordingly. In mercury the penetration is 5 times as great, and 

 the resulting expansion, therefore, about the same per unit of 



