172 Lord Rayleigh on Stresses in Solid Bodies due to unequal 



shall ensure that §Fzdz = 0. Otherwise the edges of the 

 plate could not he regarded as free from imposed stress in 

 the form of a force or couple. 



The assumption in (1), (2) thatw = v = is now replaced by 



u= (a + £*)*, v = (a + /3z)y, . . . (12) 

 and 



w=«f-i0(*»+y)j ..... (i2') 



where w' is a function of z only. We find 



7 / 



R=(\ + 2^)-^-+2X(« + £s)- 7 0, • • (13) 



P = Q = xJ / +(2X + 2ri(«+^)-y^ (14) 



S = T = U = (15) 



Since R is supposed to vanish, we get 



P== Q = 2^rf±£?-*l (16) 



In (16) a and /9 are to be determined by the conditions 

 §Pdz = 9 JPs<Z* = 0; 



or, which comes to the same, we are to reject from 6 such 

 linear terms as will leave 



J0<fe = O, J<9^2=0 (17) 



Since uJ and 6 are independent of # and y, the equations of 

 equilibrium (5) are satisfied. 



It is of interest to trace the influence of time upon the 

 double refraction of the heated plate when light passes through 

 it edgeways, e. g. parallel to y. Initially 6 may be supposed 

 to be an arbitrary function of z, while the faces of the plate, 

 say at and c, are maintained at given temperatures. Ulti- 

 mately the distribution of temperature is expressed by a 

 linear function of z 9 say H'-fK^; and, as is known from 

 Fourier's theory, the distribution at time t may be expressed 



by 



d^R' + Kz + SAne-Pntsm 7 ^, .... (18) 



where n is an integer and p n , depending also upon the con- 

 ductivity, is proportional to n 2 . After a moderate interval 



