SECT. I.] DISTRIBUTION OF HEAT IX THE RING. 221 



5th. If we wish to ascertain how much heat flows in a given 

 time, across a definite section of the ring, we must employ the 



integral - KS I dt -=- , writing for -y- the value of that function, 

 J dx cLx 



taken at the point in question. 



245. Heat tends to be distributed in the ring according to 

 a law which ought to be noticed. The more the time which 

 has elapsed increases the smaller do the terms which compose 

 the value of v in equation (E) become with respect to those 

 which precede them. There is therefore a certain value of t for 

 which the movement of heat begins to be represented sensibly 

 by the equation 



/ x x \ _Tct 



u = a n + (a. sin - 4- Z&amp;gt; cos - ) e r- . 



\ l r rj 



The same relation continues to exist during the infinite time 

 of the cooling. In this state, if we choose two points of the ring 

 situated at the ends of the same diameter, and represent their 

 respective distances from the origin by x v and # 2 , and their cor 

 responding temperatures at the time t by v l and v z ; we shall have 



Vl = Ja + (^ sin^-l-^ cos^-H e ~^^~ ht 



f , t - x * , T, X 2\ - 

 v ~ 1 a o + a i sm + &i cos e 



2 ( \ l r rj 



The sines of the two arcs and -f differ only in sign ; the 



or TT 



same is the case with the quantities cos and cos ; hence 



r r 



thus the half-sum of the temperatures at opposite points gives 

 a quantity a e~ ht , which would remain the same if we chose two 

 points situated at the ends of another diameter. The quantity 

 a e~ ht , as we have seen above, is the value of the mean tempera 

 ture after the time t. Hence the half-sum of the temperature 

 at any two opposite points decreases continually with the mean 

 temperature of the ring, and represents its value without sensible 

 error, after the cooling has lasted for a certain time. Let us 



