THE MECHANICAL INTERPRETATION OF JOINTS 5 
at the point P. When the principal stresses are unequal the nota- 
tion has been chosen so that ¢,<0,<o;. 
We now choose positive directions on the principal axes, and 
denote the positive principal radii by 7,, r,, r,. Then, if p is the 
stress at any surface 7, we have from (1) 
p COS (pry) =o, COS (77,), 
p COS (pry) =a, COS (ry), (5) 
p COS (prz) =o; COS (r7,). 
The components of the stress at any surface r are thus given in terms 
of the principal stresses and the direction cosines of r. From 
these components we may compute the normal stress, o, at 7: 
— g=p cos (pr) =a; Cos? (77_)-+oy COs? (rry)- oz COS? (772). (6) 
The equations (5) show, moreover, that the stresses over the sphere 
are symmetrically distributed with respect to each of the three 
principal planes. It is sufficient, therefore, to know the stress 
distribution in one of the octants into which the principal planes 
divide the sphere. 
We shall now examine the stress distribution in the principal 
planes. Let 7 be a radius in the «z plane making an angle ¢ =(rr,) 
with the positive z-axis. Since (7r,)=90°, we have from (5) that 
(pry) =90°; hence the stresses in a principal plane lie entirely in 
that plane. From (6) we have for the normal stress at r 
i : Cx1-Oz 
o=o, sin? ¢+a, cos? d= = 
ee cos 2g. (7) 
MomontainepaweNpUto,.—o., T2—1—O, %—7, Im (2)) then, by, 
virtue of (7), 
WEA COS p _ oy j _ U2 Gx | 
T=(¢,—c) oes (o,—o,) sind cosh a 2g. (8) 
If o, 7 are regarded as rectangular co-ordinates, equations (7) and 
(8) are the parametric equations of a circle of center [(¢,+¢,)/2, 0] 
and radius (¢,—o,)/2. We thus have the following graphical 
representation of the stress distribution in the xz plane: Lay off 
