PLANE STRAIN IN BIPOLAR CO-ORDINATES. 
287 
This has a maximum tension at the symmetrical point (x = 0) of magnitude 
4P ^/(cP—r 2 ) .(62) 
At the points x — + \/(d 2 —r i ) it vanishes, and then becomes a compression which 
reaches a maximum value at points at distances ± x /3(d 2 —r 2 ) on either side of the 
foot of the perpendicular from the centre of the 
hole, which is numerically equal to one-eighth of 
maximum tension. 
The stress round the circular hole may be 
represented by a simple geometrical construction. 
If in fig. 3 the centre of the circular hole is C, Q 
is any point on the circle, and CA the perpen¬ 
dicular drawn from C to the straight edge, and if 
Q denote the angle QAC, we easily see that 
tan <p = sin /3 cosech a u 
and the stress round the circular hole is 
/3/3 = Pj (1 + 2 tan 2 <p). . 
(63) 
Fig.3. 
Hence the stress is the same at points Q, Q' which lie on the same ray through A. 
The stress is minimum at the points nearest to and most remote from the straight 
edge, where it is a tension P numerically equal to the applied pressure. Thus at 
these points the stress is the same as it would be in the absence of the straight 
boundary if the plate were infinite. The maximum stresses are at the points of 
contact of the tangents drawn from A the circular boundary. At these points its 
value is 
Pi 
d 2 + r 2 
d 2 —r 2 
(64) 
The maximum tension in the circular boundary is equal to the maximum tension in 
the straight edge if d = 3 r. In this case each is equal to 2P 2 . If the distance of 
the hole from the straight edge is greater than this value the maximum tension is at 
a point on the circular boundary ; and if it is less, the maximum stress tension is at 
the symmetrical point on the straight edge. On the other hand, the point of 
maximum difference of principal stresses is on the straight edge or the circular 
boundary, according as d is greater or less than 2r. This suggests a simple 
method of determining whether, for a particular material, rupture occurs at the point 
of greatest tension or at the point of greatest stress-difference. If a circular hole is 
bored near the straight edge of a uniform plate, so that the distance of its centre 
from the edge is greater than ^/2 and less than ^/3 of the radius, and a uniform 
radial pressure is exerted over the hole in any convenient way and increased until 
