THE EQUILIBRIUM OF ELASTIC SOLIDS. 35 



The quantities 8x, Sy, Sz, represent the absolute motion of each point in 

 the directions of the three co-ordinates ; but as compression depends not on 

 absolute, but on relative displacement, we have to consider only the nine 

 quantities 



Since the number of these quantities is nine, if nine other independent 

 quantities of the same kind can be found, the one set may be found in terms 

 of the other. The quantities which we shall assume for this purpose are 



1. Three compressions, -, -jr, , in the directions of three principal 

 axes a, ft, y. 



2. The nine direction-cosines of these axes, with the six connecting equa- 

 tions, leaving three independent quantities. (See Gregory's Solid Geometry.) 



3. The small angles of rotation of this system of axes about the axes of 

 x, y, z. 



The cosines of the angles which the axes of x, y, z make with those of 

 a, ft, y are 



cos (aOx) a, , cos (ftOx) = ^ , cos (yOx) = c l , 

 cos (aOy) = a a , cos (ftOy) = b s , cos (yOy) = c 2) 

 cos (aOz) =a s , cos (/JOz) = b a , cos (yOz) = c 3 . 



These direction-cosines are connected by the six equations, 



t , , 3 2 c 3 = 0, 



3 2 + &/ + c s 2 = 1, ,, + &A + c 3 c, = 0. 



The rotation of the system of axes a, ft, y, round the axis of 



x, from y to z, =S# 1( 

 y, from z to x, =&&.,, 

 z, from x to y, =80.,; 



52 



