122 LOUIS VESSOT KING 
a considerable bulge of the equatorial diameter, showing that the 
problem is not strictly one of plane stress. The complete solution 
of the problem would be of the nature of a similar one considered 
by Filon* in which the state of stress is determined for a cylinder 
pressed between planes which are in contact with its ends. For 
the purposes of the present paper, however, we shall deal with the 
problem as one of plane stress and neglect surface tractions, over 
the boundary of the cylindrical specimen due to relative displace- 
ments of the rock surface and the steel jacket when under pressure. 
The results will represent to a first approximation the state of stress 
existing in the rock specimen. From the general solution of the 
problem given by Love? it is an easy matter to calculate the stress- 
differences at any point of the specimen. According to the usual 
notation we denote by #7, 66, and 22 the radial, transverse, and axial 
stresses respectively: for a case of plane stress there are also the 
principal stresses. We denote the differences of the principal 
stresses by the notation 
(i) 7#-3 (ii) 77-60 (ili) 69-28 
From the theorem cited in §1 it is seen that each of the stress- 
differences (i), (ii), (iii) is associated with a family of surfaces of 
maximum shear along which the material will crack or flow. 
(i) The surfaces of maximum principal shear due to the stress- 
differences (77—2Z2) consist of two systems of cones of semi-vertical 
angle 45° and cutting each other orthogonally. One such system 
may be imagined to consist of a pile of glass funnels fitting into 
one another. 
(ii) The surfaces of shear due to the stress-differences (77—66) 
consist of two systems of mutually orthogonal cylindrical surfaces 
whose traces on a plane perpendicular to the axis of the cylinder 
are equiangular spirals cutting the radii at angles of 45°. 
(iii) The stress-differences (@6—£2) give rise to two families of 
helicoidal surfaces of pitch 45° and intersecting orthogonally. The 
traces of these surfaces on the surface of the cylinder give rise to 
helices well known as Luder’s lines. 
tL. N. G. Filon, ‘‘On the Elastic Equilibrium of Circular Cylinders under Certain 
Practical Systems of Load,” Phil. Trans. Roy. Soc., CXCVIII A, 1902, 182. 
2 Love, Elasticity, 2d ed., 140. 
