SECT. I.] 



COMPLETE SOLUTION. 



217 



stant state in which the difference of the temperature u from the 

 constant b is represented by 



x 



x\ 

 - ) 



a sm - + b cos - e 

 r rj 



Thus the particular values which we have previously considered, 

 and from which we have composed the general value, derive their 

 origin from the problem itself. Each of them represents an 

 elementary state which could exist of itself as soon as it is sup 

 posed to be formed ; these values have a natural and necessary 

 relation with the physical properties of heat. 



To determine the coefficients , a lt a 2 , &c., 6 , 6 1? & 2 , &c., we 

 must employ equation (II), Art. 234, which was proved in the 

 last section of the previous Chapter. 



Let the whole abscissa denoted by X in this equation be 2?rr, 

 let x be the variable abscissa, and let f(x] represent the initial 

 state of the ring, the integrals must be taken from x = to 

 x = 2-Trr ; we have then 



*) ~ 3 //( 



* 



+ sin 



in (3/ si 



sn 



Knowing in this manner the values of a , 1 , a 2 , &c., 

 b , b t , b 2 , &c., if they be substituted in the equation we have 

 the following equation, which contains the complete solution of 

 the problem : 



irrv 



. x 



sm - 

 r 



COS- 



kt 



x r / 2t 



sin 2 - I ( sin / (x) a 

 rj\ r * 



cos 2 - ( fcoB /(a?) dx J 



+ &al 



(E). 



