plate were kept at lixeil temperatures, is conipensated by bending and entails no stress. In sucli cases before calculating tlie stress by (11) we must throw out the mean value of ') so as to make / P dz = 0, and also such a term proportional to the distauce from the middle surface as shall ensure that fPzdz = 0. Othenvise the edges of the plate could not be regarded as free from imposed stress in the form of a forcc or couple.
The assumption in (1), (2) that u = v= 0 is now replaced bv
« = («-f- (3z) ./•, — (3z) n, (12)
to = to'— ^ (3 (.i-2 +.»/"), (12)
where w' is a function of r only. \\ e tind
7f = (A + 2/»)^' + 2x(» + ^)— (13)
!>=<i = ?. ~ + (2;. + 2,«) (* + fi~) —/«,....(1 I)
S= T= U= 0 (!•»)
Since // is supposed to vanish, we get
/. + Zfx L y. J
In (1 li) ^ and ,3 are to be determined bv the conditions
fPdz = 0, fPzdz = 0';
or, which comes to the same, we are to reject trom 4 such linear ternis as will leave
J'idz= 0, f 6 zdz = 0 (17)
Since w' and 'i are independent of ,r and //, the equations of equilibriuni (5) are satistied.
lt is of interest to tracé the iniluence of time upon the doublé refraction of the heated plate when light passes through it edgeways, f. tj.