SECT. 1.] PARTIAL CHANGES OF TEMPERATURE. 223 



other part has a temperature less than the mean temperature, 

 and the defect is the same as the excess at the opposite point. 

 This symmetrical distribution of heat exists throughout the whole 

 duration of the cooling. At the two ends of the heated half, 

 two flows of heat are established in direction towards the cooled 

 half, and their effect is continually to bring each half of the 

 ring towards the mean temperature. 



246. - We may now remark that in the general equation which 

 gives the value of v, each of the terms is of the form 



x x\ - &amp;lt;&amp;gt; 



a, sin i - + b. cos i - } e l ^. 

 r r) 



We can therefore derive, with respect to each term, consequences 

 analogous to the foregoing. In fact denoting by X the distance 

 for which the coefficient 



a. sin i \- b. cos i 

 r r 



X 



is nothing, we have the equation 6. = a t tan i , and this sub 

 stitution gives, as the value of the coefficient, 



a being a constant. It follows from this that taking the point 

 whose abscissa is X as the origin of co-ordinates, and denoting 

 by u the new abscissa x X, we have, as the expression of the 

 changes of this part of the value of v, the function 



ae~ smi-e 



If this particular part of the value of v existed alone, so as to 

 make the coefficients of all the other parts nul, the state of the 

 ring would be represented by the function 



i&quot; 



ae~ ht e~ 



** . , .u\ 



r 2 Sin (l - } , 



\ rj 



and the temperature at each point would be proportional to the 

 sine of the multiple i of the distance of this point from the origin. 

 This state is analogous to that which we have already described : 



