ON THE STRESSES IN SOLID BODIES BUE TO UNEQUAL HEATING, &C. 35 



plate were kept at tixecl teinpemtures, is compeusated by beiidhig and 

 entails no stress, la sucli cases before calculating tbe stress by (11) Ave 

 inust throw out tlie meau value of ù so as to inake/Pffe = Oj and also 

 sucli a term proportional to the distance from the middle surface as 

 sball eiisure tbat f Pzd::=^0. Otberwise tbe edges of tbe plate could 

 not be regarded as free from imposed stress iu tbe forni of a force or 

 couple. 



Tbe assunij)tion in (1), [t) tbat /«^ r = is iiow replaccd by 



// = (^. + p.^').'', v = {x + r^.:),/, (12) 



and 



w = w'—i,rA-'-+.r), (12') 



Mbere iv is a fiinction of .: oidy. A\e find 



7*' = (/. + 2^)^-f-2A(;. + /3..)-7^ (18) 



P = ^^ = >. "^ + (SA + 2,a) U + /3..) - r ^^ . . . . (M-) 



S=T=U={) (15) 



Since 7/ is supposed to yauisb, we get 



P=U = 



[j.y 



■^ + /3.r 



H ....(16) 



;. + 2/y. L z 



In (l(i) y. and p are to be deterniined bj tbe conditions 



fPd,: = {), fp,:d~ = ; 



or, wbicb conies to tbe saine, we are to reject from Ù sucli linenr ternis 

 as will leave 



ffjdz^^), fô::d~ = (17) 



Since //•' and Ù are independent of ./• and j/, tbe ecpiations of eqiiililjriuin 

 (5) are satisiied. 



It is of interest to trace tlie inHuence of tiine upon tbe double refrac- 

 tion of tbe beated plate wben ligbt passes tbrougb it edgeways, e. y. 



3* 



