86 



"2" 



R = -;r- tannic, (6) 



which shews the value of R retains the same form whatever 

 the inclination of CD may be, if DE continues at right angles 

 to it, as may also be seen, in the more general case, from equa- 

 tion (2) or (5). 



When — c — 6, 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, 



R=-2-Xsin2(c-^>). (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 6 z: 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), 



R in (6) : R in (7) : : 283 : 687 : : 3 : 7, nearly. 



In fluids the ratio will be as 1 to I, and when the angle of 

 repose is 90°, the ratio will be as 1 to 4 ; for tan'^l c : sin^c : : 

 ^ c^ : 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 C O 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 arc 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. 



