CHAPTER IV. 



OF THE LINEAR AND VARIED MOVEMENT OF HEAT IN A RING. 



SECTION I. 



General solution of the problem. 



238. THE equation which expresses the movement of heat 

 in a ring has been stated in Article 105 ; it is 



dv _ K d 2 v hi ,7 N 



dt~Cl)dx*~~CDS V 



The problem is now to integrate this equation : we may 

 write it simply 



dv d*v , 



wherein k represents -= , and h represents yrrTa &amp;gt; x denotes the 



length of the arc included between a point m of the ring and the 

 origin 0, and v is the temperature which would be observed at 

 the point m after a given time t. We first assume v = e~ ht ufx 



7 72 V 



u being a new unknown, whence we deduce -ji = k T~2 now this 



equation belongs to the case in which the radiation is nul at 

 the surface, since it may be derived from the preceding equa 

 tion by making h = : we conclude from it that the different 

 points of the ring are cooled successively, by the action of the 

 medium, without this circumstance disturbing in any manner the 

 law of the distribution of the heat. 



In fact on integrating the equation -77 = &-TT &amp;gt; we should 



dt (tx 



find the values of u which correspond to different points of the 



