86 
a) 
Bia “> tan’ de, (6) 
which shews the value of R retains the same form whatever 
the inclination of CD may be, if DE continues at right angles 
toit, as may also be seen, in the more general case, from equa- 
tion (2) or (5). 
When 0 = c — 3b, DE stands at the angle of repose, and 
when infinite, from equation (4), the fracture CF becomes 
parallel to it; and from equation (2) or (5) may be had, by 
reduction, 
rpy2 
ie > x sin®(c—d). (7) 
As in loose stuff DE can never stand at a steeper inclination 
than the angle of repose, equation (7) gives the greatest value 
R can ever attain; the height of the face H being constant. 
When 4 = 0, H vertical, and c = 34°, which corresponds to 
a slope of repose of 1} to 1 nearly, we get, by comparing 
equations (6) and (7), 
Rin (6): R in (7) : : 283: 687 :: 3: 7, nearly. 
In fluids the ratio will be as 1 to 1, and when the angle of 
repose is 90°, the ratio will be as 1 to 4; for tan’}c:sin’e:: 
1¢?:c?::1:4: hence no instance can occur wherein, the face 
H being vertical, the value of R can exceed four times the 
value when the top DE is horizontal. 
Equation (4) gives the following simple geometrical con- 
struction for finding the line CF. Draw any line MO, cut- 
ting the line of repose CE at right angles in K, and termi- 
nating in the face CD at M, and in the line CO parallel to 
DE, the top, in O. On MO describe a semicircle, cutting 
CE, the line of repose, in H; from O as centre, with OH as 
radius, describe an are to cut OM in I; join C to I, and pro- 
duce CI to F; then the wedge FDC will require the maxi- 
mum resistance. When DE is at right angles to DC, the 
angle ECD is bisected by EF. 
