226 THEORY OF HEAT. [CHAP. IV. 



denly added to that of the body with which it is in contact; and a 

 common temperature results which is equal to the quotient of the 

 sum of the quantities of heat divided by the sum of the masses. 

 Let ft) be the mass of the infinitely small layer which is separated 

 from the hotter body, whose temperature is a; let a and ft be the 

 variable temperatures which correspond to the time t, and whose 

 initial values are a and Z&amp;gt;. When the layer co is separated from the 

 mass m which becomes m &&amp;gt;, it has like this mass the tempera 

 ture a, and as soon as it touches the second body affected with the 

 temperature /3, it assumes at the same time with that body a 



temperature equal to . The layer a, retaining the last 



temperature, returns to the first body whose mass is m co and 

 temperature a. We find then for the temperature after the second 

 contact 



. /w/3 + aftA 



a [m a)) + &&amp;gt; 



v \ m + co } c:m 



or 



m m 4- G) 



The variable temperatures a. and /3 become, after the interval 

 dt, a. -f (a ft} , and ft -f (a /3) ; these values are found by 



Tfl&amp;gt; f ITb 



suppressing the higher powers of co. We thus have 



the mass which had the initial temperature (3 has received in one 

 instant a quantity of heat equal to md@ or (a ft) co, which has 

 been lost in the same time by the first mass. We see by this 

 that the quantity of heat which passes in one instant from the 

 most heated body into that which is less heated, is, all other things 

 being equal, proportional to the actual difference of temperature 

 of the two bodies. The time being divided into equal intervals, 

 the infinitely small quantity co may be replaced by kdt, k being the 

 number of units of mass whose sum contains co as many times as 



the unit of time contains dt, so that we have - = We thus 



co dt 



obtain the equations 



dz = -(a-j3)~dt and d& = (a - 0) - dt. 



