228 THEORY OF HEAT. [CHAP. IV. 



&amp;lt;?=(-)-&amp;lt;#, 



m 



in which k would be equal to a function of a and /?, which we 

 denote by &amp;lt;f&amp;gt; (a, /?). It is easy to ascertain the law which 

 the variable temperatures a and /3 follow, when they approach 

 extremely near to their final state. Let y be a new unknown 

 equal to the difference between a and the final value which is 



^ (a + 6) or c. Let z be a second unknown equal to the difference 

 2 



c p. We substitute in place of a and /3 their values c y and 

 c 2 ; and, as the problem is to find the values of y and z, 

 when we suppose them very small, we need retain in the results 

 of the substitutions only the first power of y and z. We therefore 

 find the two equations, 



k 



-dy = -(z-y}^(c-y ) c-z)dt 



k 



and dz (z y] $(c y, c z) dt, 

 tail 



developing the quantities which are under the sign (/&amp;gt; and omit 

 ting the higher powers of y and z. We find dy=(z y) $&amp;gt;dt, 



and dz = (z y] &amp;lt;f&amp;gt;dt. The quantity $ being constant, it 



7?2&amp;gt; 



follows that the preceding equations give for the value of the 

 difference z y,& result similar to that which we found above for 

 the value of a /3. 



From this we conclude that if the coefficient k, which was 

 at first supposed constant, were represented by any function 

 whatever of the variable temperatures, the final changes which 

 these temperatures would experience, during an infinite time, 

 would still be subject to the same law as if the reciprocal con- 

 ducibility were constant. The problem is actually to determine 

 the laws of the propagation of heat in an indefinite number of 

 equal masses whose actual temperatures are different. 



251. Prismatic masses n in number, each of which is equal 

 to m, are supposed to be arranged in the same straight line, 

 and affected with different temperatures a, b, c, d, &c. ; infinitely 



