Thermometers with Spherical and Cylindrical Bulbs. 137 



The lag of a thermometer, whose spherical bulb is subject 

 to the conditions just considered, is therefore 



SI-SI-- 4 --' (4) 



This approaches a steady value, L, as the time t increases, 

 which is 



~15X~15a 2 ~ 15K l ° j 



The steady lag is thus determinate, for a given gradient G, 

 when X is known. We can measure \ experimentally by 

 observing the rate of cooling of the spherical bulb. For 

 suppose the temperature of the medium is zero, and the 

 initial distribution of temperature in the sphere is given 

 by v = <f>(r). We neglect the effect of the glass wall, if we 

 are dealing with a liquid-in-glass thermometer, and suppose 

 the surface conductivity to be infinite as before. Then, 

 to get the temperature in the cooling bulb at any instant, 

 we require a solution of (2) which satisfies the conditions 



v = <j)(r) for t = 0, 



v — for r = 0, 



v = for r = c = radius of sphere. 



Such a solution is (Weber, vol. ii. § 46) : 



2 <*-» 2W . nirr \ c . . . . 

 v = - > e n ' M sin I <p{a.) sin {iittoljc) da.. 



c S 6 ' ^o 



If the initial temperature is constant and equal to u , 

 we have (j>{r) — ru . On substituting, and taking the mean 

 temperature, we get, since v — ru, 



If t x is the time required for the mean temperature 

 to drop from u to fu , where / is some positive fraction, 

 we have 



/tt 2 /6 = e-"i+±e- 4J ^+£*- 9A ' 1 + (6) 



And X may be found, by successive approximations, from 

 this equation, one or two terms on the right sufficing for 

 most cases. 



