CHAP. VI.] COMPLETE SOLUTION. 309 



which gives the value of a t . The coefficients a l9 a 2 , a 3 , . . . a p being 

 thus determined, the condition relative to the initial state expressed 

 by the equation &amp;lt;f&amp;gt; (x) = a^ + a 2 u 2 + a 3 u s + &c., is fulfilled. 



We can now give the complete solution of the proposed problem; 

 it is expressed by the following equation : 



f- 

 J 







i _ 







+ &C. 



The function of a? denoted by u in the preceding equation is 

 expressed by 



all the integrals with respect to # must be taken from a? = to 

 x X, and to. find the function u wer must integrate from q = to 

 &amp;lt;2 = 7r; (a?) is the initial value of the temperature, taken in the 

 interior of the cylinder at a distance # from the axis, which 

 function is arbitrary, and 6 V 6 Z , 6 y &c. are the real and positive 

 roots of the equation 



J L X -JL JL JL _L 6 

 &quot;2 ~ F^ ^ 3 - 4^ 5-&c. 



320. If we suppose the cylinder to have been immersed for 

 an infinite time in a liquid maintained at a constant temperature, 

 the whole mass becomes equally heated, and the function (/&amp;gt; (x) 

 which represents the initial state is represented by unity. After 

 this substitution, the general equation represents exactly the 

 gradual progress of the cooling. 



If t the time elapsed is infinite, the second member contains 

 only one term, namely, that which involves the least of all the 

 roots lt 2 , V &c.; for this reason, supposing the roots to be 

 arranged according to their magnitude, and to be the least, the 

 final state of the solid is expressed by the equation 





