6 WALTER H. BUCHER 
OX =o,, OZ =o; on the axis of abscissas (Fig. 11), and draw a circle 
having XZ as diameter. This circle has the center and radius speci- 
fied above; and the abscissa and ordinate of any point R of the circle 
represent the normal and shearing stress, o, 7, respectively, which 
correspond to a radius 7 of the sphere in the xz plane and making 
an angle ¢@ with 7, equal to one-half of the central angle ZCR. 
If we lay off OY =a,, a similar argument shows that the stress 
distributions on the xy and yz planes are represented by the 
co-ordinates of the points forming the circles on XY and VZ as 
Fic. 11 
diameters. It may also be shown that for any radius of the sphere 
not in the principal planes, the corresponding stresses oc, 7, are 
represented by the co-ordinates of points within the region bounded 
by all three of the circles XY, YZ, XZ. The circle XZ therefore 
passes through all the points which have the greatest r for any given o 
occurring in the stress distribution. ‘This circle plays an important 
role in Mohr’s theory of rupture, and is called by Mohr the principal 
circle in the graphic representation of the state of stress. 
Mohr’s theory of rupture deals with cases in which the failure 
of the material is assumed to be due to the sliding of one layer over 
another in certain planes called shearing planes. The fundamental 
