36 



THE EQUILIBRIUM OF ELASTIC SOLIDS. 



By resolving the displacements Sa, 8)8, Sy, 80!, 80,, 80,, in the directions 

 of the axes x, y, z, the displacements in these axes are found to be 



Bx = ,8a + 6,8/J + c,8y 80,z + 80,y , 

 8y = a,8a + 6,8)3 + c,8y 80,x + 80,2, 

 Bz = a,8a + 6,8)8 + c,Sy - S0,?/ + 80,*. 



a B y 



and a = a,z + a,y + a,z, B = b^c + bj/ + bf, and y = c,x + c^ + c,z. 



Substituting these values of 8a, 8)8, and 8y in the expressions for Bx, Sy, 

 82, and differentiating with respect to x, y, and z, in each equation, we obtain 

 the equations 



dBx _ Ba , 8/8 , , 8y , 

 dx a 1 B 1 y ' 



dSy Sa 8/8 . . Sy 

 -,' / =--a 1 &'-c 1 



Sa 



y 



. + fe. 



x Sa 8)8 7 7 Sy , 



= = a A + 6i&> + CA " 



_ So, 8^8 Sy 



~ " \JvtfJL* ^" ^~ ^2^1 "T "' Q^\ """" ^ 



* Aj w 



rt C* 



op 7 7 oy 

 p y 



= 8a a 8jt bb Sy 



Sa 

 a 



Sa 

 a 



Sa 



.(1). 



Equations of 

 compression. 



(2). 



Equations of the eqi'i;/,n'>im of an element of the solid. 

 The forces which may act on a particle of the solid are : 



1. Three attractions in the direction of the axes, represented by X, Y, Z. 



2. Six pressures on the six faces. 



