270 



K. YAMAGAWA 



and Di is lo be determined l)y 



Hence the general solution is 

 ] -1^)' = «) 2(iX 



V = 



^ ^/ = i in ^^-WX + ^ 2,,iX _ , ,,,o„ «x 1 V ^ >* 



This solution, however, is only true, when the c^'cliial process of 

 heating and cooling has been repeated for an intinite luuubcr of times, 

 so as to ettace completely the trace of the initial distribution of tem- 

 perature. If by repetition of the cyclical process, the initial effect be 

 completely destroyed, then the temjieraturc distribution would depend 

 on the surface condition only. Ihit the solution satisfies the surface 

 condition, and the ditferential equation ; therefore the value of it cor- 

 responding t(j any valne of t and x must give the temperature at x 

 and at t. Hence the solution is unique. 



When X is made ecpial to zero, the value of « will rcjiresent the 

 temperature at the centre. Denoting by m,,) the value of « for x—O^ 

 we obtain 



4rt s:^i = <^ Pr X 



%,= 2j . = 



\.:..{^..,>) 



"" ^ = 1 i v/ £-- '■'■' ^ + £-'■'' ^ ^ 2 COS 2 y ,.' X 



TT 



or changing l>i to ß^— -—- , 



which might br written 



u„ = ai sm (-^ t 4- /3, j + a^ sin (■'> ^ t + ß-^) + &c. 



