THEORY OF HEAT. [CHAP. IV. 



If the second mass alone had been heated and the tempera 

 tures ,, a 3 , 4 , ... a n were nul, we should have 



2 + - a 2 jsin (j - 1) sin 



2vr 2?r) 

 + cos (/I) cos ^e &quot; 

 Vl/ 7 n w I 



Bin2 



n 



^ 



+ cos (7 -1)2 cos 2 

 Vi/ 



4-&C., 



and if all the initial temperatures were supposed nul, except 

 t and a 2 , we should find for the value of a j the sum of the values 

 found in each of the two preceding hypotheses. In general it is 

 easy to conclude from the general equation (e), Art. 273, that in 

 order to find the law according to which the initial quantities of 

 heat are distributed between the masses, we may consider sepa 

 rately the cases in which the initial temperatures are mil, one only 

 excepted. The quantity of heat contained in one of the masses 

 may be supposed to communicate itself to all the others, regarding 

 the latter as affected with nul temperatures; and having made 

 this hypothesis for each particular mass with respect to the initial 

 heat which it has received, we can ascertain the temperature of 

 any one of the bodies, after a given time, by adding all the 

 temperatures which the same body ought to have received on 

 each of the foregoing hypotheses. 



275. If in the general equation (e) which gives the value of 

 a jt we suppose the time to be infinite, we find a,- = - 2 a i} so that 



each of the masses has, acquired the mean temperature ; a result 

 which is self-evident. 



As the value of the time increases, the first term - 2 &i 



n 



becomes greater and greater relatively to the following terms, or 

 to their sum. The same is the case with the second with respect 

 to the terms which follow it; and, when the time has become 



