218 THEORY OF HEAT. [CHAP. IV. 



All the integrals must be taken from x = to x = 2?rr. 



The first term ^ \f( x ] d x &amp;gt; which serves to form the value of 



v, is evidently the mean initial temperature, that is to say, that 

 .which each point would have it&quot; all the initial heat were distri 

 buted equally throughout. 



242. The preceding equation (E) may be applied, whatever 

 the form of the given function f(x) may be. We shall consider 

 two particular cases, namely : 1st, that which occurs when the 

 ring having been raised by the action of a source of heat to its 

 permanent temperatures, the source is suddenly suppressed ; 2nd, 

 the case in which half the ring, having been equally heated 

 throughout, is suddenly joined to the other half, throughout which 

 the initial temperature is 0. 



k 1 



We have seen previously that the permanent temperatures 



of the ring are expressed by the equation v = az x + bz~ x ; the 



value of quantity a being e KS , where I is the perimeter of the 

 generating section, and S the area of that section. 



If it be supposed that there is but a single source of heat, the 

 equation -7- = must necessarily hold at the point opposite to 



that which is occupied by the source. The condition aoL x boT x = 

 will therefore be satisfied at this point. For convenience of calcu 

 lation let us consider the fraction -yj to be equal to unity, and let 



us take the radius r of the ring to be the radius of the trigono 

 metrical tables, we shall then have v = ae x + be~ x ; hence th&amp;lt;~mitial 

 state of the ring is represented by the equation 



v = le*(e*+*+e). 



It remains only to apply the general equation (E), and de 

 noting by M the mean initial heat (Art. 241), we shall have 



This equation expresses the variable state of a solid ring, which 

 having been heated at one of its points and raised to stationary 



