
EQUILIBRIUM OF ELASTIC SOLIDS. 91 





dox doz doz 
dz dy dz 
doy doy doy 
dx dy dz 
doz doz doz 
dx dy dz 
Since the number of these quantities is nine, if nine other independent quan- 
tities 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, = ae oy, in the directions of three principal axes 
Bean: Bey, 
2. The nine direction-cosines of these axes, with the six connecting equations, 
leaving three independent quantities. (See Grecory’s Solid Geometry). 
3. The small angles of rotation of this system of axes about the axes of 4, y, 2, 
The cosines of the angles which the axes of x, y, make with those of a, 8, y 
are— 
cos (20 x)=a,, cos (8 0 x) =b,, cos (y 02) =c,, 
cos (a0 y)=a,, cos (8 0 y)=6,, cos (y 0 y)=c., 
cos (a 0 z)=a,, cos (6 0 z)=6,, cos (y 0 z)=c;, 
These direction-cosines are connected by the six equations, 
a?+b?+e7=1 a, a+b, b,+¢, c,=0 
a, +0?,+¢72=1 Gy A + b,'b, + €, C,=0 
a2 + b,? + ¢,7=1 a, a, +b, b, +¢, ¢,=0 
The rotation of the system of axes a, 8, y, round the axis of 
x, from y to z, =0 6, 
y, from z to x, =0 6,, 
z, from x to y, =0 0,; 
By resolving the displacements 6 a, 6 6, dy, 6,, 6,, 6, in the directions of the 
axes 2, y, 2, the displacements in these axes are found to be 
da=a,0a+b, 0B +e,d0y— 6,2 + Oy 
Oy =a,6a+b,08 +e,0y—- 6,244 6,2 
d2=a,da+b,0B+e,0y7— Oy + O% 
aye 6B oy 
But ba—a--, 0f=8-G> and dy=y-—. 
and a=a,2+a,y+a,2, B=b,7+b,y+6,2, and y=e,x+e,y+¢,¢. 
Substituting these values of 6 a, 68, and dy in the expressions for dz, dy, 
VOL. XX. PART I. 2B 
