222 THEORY OF HEAT. [CHAP. IV. 



examine more particularly in what the final state consists, which 

 is expressed by the equation 



f / X -L X \ --} M 



v = -\a Q + f j sin - + 6, cos -j e &amp;lt;*&amp;gt; e~ ht . 



If first we seek the point of the ring at which we have the 

 condition 



/7i \ 



a, sin - + b cos - = 0, or - = arc tan ( ) , 

 r r r \aj 



we see that the temperature at this point is at every instant 

 the mean temperature of the ring : the same is the case with 

 the point diametrically opposite ; for the abscissa x of the latter 

 point will also satisfy the above equation 



IT f r) 



- = arc tan I L 



r \ a^ 



Let us denote by X the distance at which the first of these 

 points is situated, and we shall have 



X 



sin 



* = - a y; 



cos 

 r 



and substituting this value of b lt we have 



cos 

 r 



If we now take as origin of abscissae the point which corre 

 sponds to the abscissa X, and if we denote by u the new abscissa 

 x X, we shall have 



= e~ ht a + sn - e 



At .the origin, where the abscissa u is 0, and at the opposite 

 point, the temperature v is always equal to the mean tempera 

 ture ; these two points divide the circumference of the ring into 

 two parts whose state is similar, but of opposite sign ; each point 

 of one of these parts has a temperature which exceeds the mean 

 temperature, and the amount of that excess is proportional to 

 the sine of the distance from the origin. Each point of the 



