SECT. II.] EQUAL PRISMATIC MASSES IN LINE. 229 



thin layers, each of which has a mass co, are supposed to be 

 separated from the different bodies except the last, and are. 

 conveyed in the same time from the first to the second, from 

 the second to the third, from the third to the fourth, and so 

 on ; immediately after contact, these layers return to the masses 

 from which they were separated ; the double movement taking , 

 place as many times as there are infinitely small instants dt\ it I 

 is required to find the law to which the changes of temperature r - 

 are subject. 



Let a, {$,%$,... co, be the variable values which correspond to 

 the same time t, and which have succeeded to the initial values 

 a, b, c, d, &c. When the layers co have been separated from the 

 n 1 first masses, and put in contact with the neighbouring 

 masses, it is easy to see that the temperatures become 



a(m co) ft (m co) -f aco 7 (m co) + {3co 

 m o) m m 



S (m co) + 70) ma) 



m m + co 



or, 



a, /3 + (a-/3)^, 7+ (-7)^, * + (7- 8)^, ... 



When the layers co have returned to their former places, 

 we find new temperatures according to the same rule, which 

 consists in dividing the sum of the quantities of heat by the sum 

 of the masses, and we have as the values of a, ft, 7, S, &c., after 

 the instant dt, 



7+ - 7- 7) &amp;gt; &quot;&amp;gt; + (f - &amp;gt;) 



The coefficient of is the difference of two consecutive dif- 

 m 



ferences taken in the succession a, /5, 7, ... -^, co. As to the first 

 and last coefficients of , they may be considered also as dif 



ferences of the second order. It is sufficient to suppose the term 

 a to be preceded by a term equal to a, and the term co to be 



