4. WALTER H. BUCHER 
Substituting the values in (1) and noting the above values for oy, 
G2, Tx, T2, We have 
ox COS (772) +7: Sin (772) =o, COS (7x72) —Tz Sin (7,72), 
or 
(o.—o;) COS (7472) =(7,+72) sin (7472). (2) 
As a first application of this equation we have 
Tz+T2=0 when (%72)=00. (3) 
In this case the shearing stresses are numerically equal but opposite 
in sign. 
Next, let 7, be any fixed radius in the plane 7,, r.; and write 
b= (rors), b+ AG=(For2)} C2=01+Ac, T2=71+Ar. 
Then (2) assumes the form 
Ao cos Ad = (27,;+Ar) sin Ad; 
dividing through by A¢ and letting A@ approach zero, we obtain 
ant (a 
the subscript being no longer needed. From this equation it 
appears that o increases with ¢@ when 7 has the direction of increas- 
ing . 
If ¢ is not constant over the sphere it must reach a minimum o, 
and a maximum go, at certain diameters which we denote by x 
and zg respectively. At these points = =o for al] diametral planes; 
hence from (4) the fofal shear vanishes at x and g. Also, from (2), 
cos (xz) =o, so that x and z are perpendicular. Again, if y is a third 
diameter, perpendicular to both x and zg, the components of shear 
in both yx and yz planes vanish by virtue of (3), and hence the 
total shear for y is also zero. The three mutually perpendicular 
diameters x, y, z, are called the principal axes, the planes xy, yz, 2x, 
the principal planes, and the stresses oy, Fy, o;, the principal stresses 
