ON THE STRESSES IN SOLID BODIES DUE TO UNEQUAL HEATINGj &C. 37 



if we suppose further that tlie initial température is uniform and 

 equal to (r), we tind bij Fourier's methoJs 



.7, =-(0-i/), .l, = ~{e-]l), J,=^{e-B),... (22) 



and 



TT Ô' ,/ îT^' 2\ , , ,/ Sttz , 2 \ 

 -1 f7 = <?~^'' ( cas ) — ke i'-''{ cos h-— 



'OTT, 



-U 1 e -i'^-' [cas 



^ G 5' 



where also 



lh='^ïh, lh = ^-^lh, ^•'^ (2-1-) 



At the middle surface^ where „ = 0^ the riglit-liand member of (23) 

 becomes 



\ / 



Initiale 



77 



'? —• i 



4 .■¥ = »• 



as was required. If we put e~'''' = T, (25) may be written 



and (26) may bc tabulated as a functiou of T, and thence of /. Il vanis- 

 hes when T=\ and wlien 7'^0. The maximum value occurs when 

 7'=*747. When 7' is less than tliis, wliieh corresponds to aii increased 

 value of t, only the iirst two or three terms iu (26) need be regarded. 

 The above value of 7' gives 



p,f= -292, 



