282 THEORY OF HEAT. [CHAP. V. 



Thus the time of cooling iu bodies of small dimension does not 

 depend on the interior conducibility ; and the observation of these 

 times can teach us nothing about the latter property ; but it 

 could be determined by measuring the times of cooling in vessels 

 of different thicknesses. 



298. What we have said above on the cooling of a sphere 

 of small dimension, applies to the movement of heat in a thermo 

 meter surrounded by air or fluid. We shall add the following 

 remarks on the use of these instruments. 



Suppose a mercurial thermometer to be dipped into a vessel 

 filled with hot water, and that the vessel is being cooled freely 

 in air at constant temperature. It is required to find the law 

 of the successive falls of temperature of the thermometer. 



If the temperature of the fluid were constant, and the thermo 

 meter dipped in it, its temperature would change, approaching 

 very quickly that of the fluid. Let v be the variable temperature 

 indicated by the thermometer, that is to say, its elevation above 

 the temperature of the air ; let u be the elevation of temperature 

 of the fluid above that of the air, and t the time corresponding 

 to these two values v and u. At the beginning of the instant 

 dt which is about to elapse, the difference of the temperature 

 of the thermometer from that of the fluid being v u, the variable 

 v tends to diminish and will lose in the instant dt a quantity 

 proportional to v u ; so that we have the equation 



dv = li (v u) dt. 



During the same instant dt the variable u tends to diminish, 

 and it loses a quantity proportional to u, so that we have the 

 equation 



du = Hudt. 



The coefficient H expresses the velocity of the cooling of the 

 liquid in air, a quantity which may easily be discovered by ex 

 periment, and the coefficient h expresses the velocity with which 

 the thermometer cools in the liquid. The latter velocity is very 

 much greater than H. Similarly we may from experiment 

 find the coefficient h by making the thermometer cool in fluid 

 maintained at a constant temperature. The two equations 



du = Hudt and dv = h (v u) dt, 



