Thermometers with Spherical and Cylindrical Bulbs. 135 



of the medium is, when steady, of the form MGyoc/K, where 

 G is the constant rate of change of temperature of the 

 medium, p is the density, a the specific heat, and K 

 the thermal conductivity of the substance in the thermo- 

 meter bulb. M is a numerical constant whose value 

 depends on the shape and dimensions of the bulb. 



It should be noted that for a given dilatation of the 

 liquid filling the bulb, the lag in a uniform gradient varies 

 almost inversely as the conductivity, since the product pa 

 is approximately constant for most substances used in 

 thermometers. 



Infinite surface conductivity implies, in the case of a 

 cooling bulb, that the medium can carry away heat from the 

 surface as fast as or faster than it can arrive there. If the 

 fluid medium in which the bulb is situated has a low con- 

 ductivity, as is the case with air, this can only be attained 

 by a sufficiently rapid movement of the medium past the 

 bulb. For air, this implies a much greater velocity relative 

 to the surface of the bulb than for liquids (of the order of 

 3000 times for considerable temperature gradients in the 

 bulb). When the surface conductivity is finite, there is a 

 definite, discontinuous change in the temperature at the 

 surface of the bulb. The expression for this temperature 

 difference is of the form NGrp<j//i, where h is the surface 

 conductivity, and N is a constant of the same nature as M. 



Radiation, and convection in the liquid in the bulb of the 

 thermometer, are neglected ; and no attempt has been made 

 to correct for the exposed stem or for the glass wall of the 

 bulb. 



Numerical results are given for mercury and alcohol, in 

 spherical and cylindrical bulbs, on a descending aeroplane. 

 The cylindrical bulbs may, in practice, be wound in the 

 form of a flat spiral.- These calculated results show that 

 for spherical bulbs of mercury and alcohol giving the same 

 volume expansion, the lags are about the same. 



With the same thermometric substance, the lag is less for 

 cylinders 10 cm. long than for spheres giving the same 

 volume expansion, and it is less for a cylinder of alcohol 

 than for a cylinder of mercury. 



1. Lag for a Sphere with Variable Surface Temperature 

 and Infinite Surface Conductivity. 



Let u be the temperature at a distance r from the centre 

 of the sphere, and let t be the time. The equation to be 



