214 THEORY OF HEAT. [CHAP. IV. 



ring at the same instant, and we should ascertain what the state 

 of the solid would be if heat were propagated in it without any 

 loss at the surface ; to determine then what would be the state 

 of the solid at the same instant if this loss had occurred, it will 

 be sufficient to multiply all the values of u taken at different 

 points, at the same instant, by the same fraction e~ ht . Thus the 

 cooling which is effected at the surface does not change the law 

 of the distribution of heat ; the only result is that the tempera 

 ture of each point is less than it would have been without this 

 circumstance, and the temperature diminishes from this cause 

 according to the successive powers of the fraction e~ ht . 



239. The problem being reduced to the integration of the 



7 72 



equation -j- = k , 2 , we shall, in the first place, select the sim- 

 dt dx* 



plest particular values which can be attributed to the variable 

 u ; from them we shall then compose a general value, and we 

 shall prove that this value is as extensive as the integral, which 

 contains an arbitrary function of or, or rather that it is this 

 integral itself, arranged under the form which the problem re 

 quires, so that there cannot be any different solution. 



It may be remarked first, that the equation is satisfied if we 

 give to u the particular value ae mt sin nx, m and n being subject 

 to the condition m Jen*. Take then as a particular value of 

 u the function e~ knH sin nx. 



In order that this value may belong to the problem, it must 

 not change when the distance x is increased by the quantity 2?rr, 

 r denoting the mean radius of the ring. Hence Zirnr must be a 



ft 



multiple i of the circumference 2?r ; which gives n = - . 



We may take i to be any integer; we suppose it to be 

 always positive, since, if it were negative, it would suffice to 

 change the sign of the coefficient a in the value ae~ knH sin nx. 



_ k n fa 

 The particular value ae r * sin could not satisfy the problem 



proposed unless it represented the initial state of the solid. Now 



7 or 

 on making t = 0, we find u = a sin : suppose then that the 



