CHAP. VIII.] ONE SOLUTION ONLY. 327 



n l} w 2 , ?i g , &c. being given by the following equation 



ha 



in which e represents na and the value of /x, is 





2 V 2n^a } 



In the same manner the functions &amp;lt;f&amp;gt; (y y t), $ (z, t) are found. 



337. We may be assured that this value of v solves the pro 

 blem in all its extent, and that the complete integral of the partial 

 differential equation (a) must necessarily take this form in order 

 to express the variable temperatures of the solid. 



In fact, the expression for v satisfies the equation (a) and the 

 conditions relative to the surface. Hence the variations of tempe 

 rature which result in one instant from the action of the molecules 

 and from the &quot;action of the air on the surface, are those which we 

 should find by differentiating the value of v with respect to the 

 time t. It follows that if, at the beginning of any instant, the 

 function v represents the system of temperatures, it will still 

 represent those which hold at the commencement of the following 

 instant, and it may be proved in the same manner that the vari 

 able state of the solid is always expressed by the function v, in 

 which the value of t continually increases. Now this function 

 agrees with the initial state: hence it represents all the later 

 states of the solid. Thus it is certain that any solution which 

 gives for v a function different from the preceding must be wrong. 



338. If we suppose the time t, which has elapsed, to have 

 become very great, we no longer have to consider any but the 

 first term of the expression for v ; for the values n v n^ n 3 , &c. are 

 arranged in order beginning with the least. This term is given 

 by the equation 



/sin ?? 1 a\ 5 

 v = ( -) cos n^x cos n^y cos n^z 



this then is the principal state towards which the system of tem 

 peratures continually tends, and with which it coincides without 

 sensible error after a certain value of t. In this state the tempe- 



