236 THEORY OF HEAT. [CHAP. IV. 



The quantities a^ b l} q, d lt and C being determined by these 

 equations, we know completely the values of the variables 



a, 0, 7, 3, ...co. 



We can in general effect the elimination of the unknowns in 

 these equations, and determine the values of the quantities 

 a, b, c, d, &c. ; even when the number of equations is infinite ; we 

 shall employ this process of elimination in the following articles. 



256. On examining the equations which give the general 

 values of the variables a, j3, 7 ...... o&amp;gt;, we see that as the time 



increases the successive terms in the value of each variable de 

 crease very unequally : for the values of u, u, u&quot;, u&quot;, &c. being 



- 7T 7T 7T , 7T p 



1-, 2-, 3-, 4 - , &c., 

 n n n n 



the exponents versin u, versin u, versin u&quot;, versin u&quot;, &c. 

 become greater and greater. If we suppose the time t to be 

 infinite, the first term of each value alone exists, and the tempera 

 ture of each of the masses becomes equal to the mean tempera 



ture - (a + b + c +...&G.). Since the time t continually increases, 



IV 



each of the terms of the value of one of the variables diminishes 

 proportionally to the successive powers of a fraction which, for the 



2fc 2Jfc 



versin u -- versin u 



second term, is e &quot; , for the third term e n , and so on. 

 The greatest of these fractions being that which corresponds to 

 the least of the values of u, it follows that to ascertain the law 

 which the ultimate changes of temperature follow, we need con 

 sider only the two first terms; all the others becoming incom 

 parably smaller according as the time t increases. The ultimate 

 variations of the temperatures a, ft, 7, &c. are therefore expressed 

 by the following equations : 



1 , 1 . Sin U - Sin Qu -versinu 



a = - (a + b + c + &c. + a 



n sin u 



1 f - - , Sill 2lt Sin U ~*~ versin 



P=-(a + + c + &c.) + &amp;lt;LI - : -- e m 



n ^ sin u 



1, 7 - . S - 



7 = - (a + b + c + &c.) + cfj 



^ 



n sm u 



