ON THE STRESSES IN SOLiD BOfilES DUE TO UNEQUAL HEATING^ &C. 38 



ked^ t believe^ by J. lIoficiNsoN '). It will be coiiveuieiit to repeat in 

 a soiuewhat dilleveut notation his formulation of tlie gênerai tlieoiy, 

 and afterwards to apply it to sonie spécial problenis to wliich tlie opti- 

 cal metliod of examination is applicable. 



In the usual notation -) if F, (i, R, S, T, U be tlie components of 

 stress; ii, v, w the displacements at tlie point .r, //, c; /., y. tlie elastic 

 constants; \ve liave sucli équations as 



/du dv , d.w\ , ^ du , , 



/dw , do\ , , 



■^=K,-?F+z-) (^) 



Thèse hold wlien the niaterial is at the standard température. If we 

 suppose tliat the température is raised by ^ and that no stresses are 

 applied, 



du dv dw 



dx dy dz ' ' 



Avhik' divldfj &c. vanish. The stresses that would be needed to ])roduce 

 the sa nie displacemcnts without change of température are 



F=Q = E = (;3a + 2,a) yj, 

 S=T= U={). 



Hence, so far as the principle of superposition liolds good^ we luay 

 Write in gênerai 



/du . dv , div\ , , du ,^ , , 



„ /dw , dv\ ^ ^ 



'^-"■h,+di} w 



with similar équations for Q, It, T, U. 



If there be no bodily forces the équation of equilibrium is 



dP , dU ^ dT ^ 



1^ + ^ + ^ = '^ ^'^ 



') Mess, of Math. vol. viii. p. 108 (1879). 



') See, for example, Love's Theory of Elasticity,' Cambridge University 

 Press, 1892. 



AKCtaVES NKEKLAXDAÎSKS, SKKIK If. TOME V. 3 



