279 



tan ^ = tan ^c; (13) 



R = ---secc^7- 7 ,) ' (14) 



2 \2 tan c + sec c J 



The value of the angle of fracture is of the same form as 

 that of Prony for a vertical face and horizontal top. 



The equations show that the stability imparted to a 

 structure at the face of a bank, by friction, arises principally 

 from the direction of the resulting force, which makes an 

 angle equal to the complement of the angle of repose with the 

 face, and that this force is in general less than the horizontal 

 force derived from the equation of Prony, or any other in which 

 face friction is neglected ; that the values of both forces, for 

 ordinary banks, are equal at angles of repose in and about 45°; 

 that the former are least for angles of repose less than this, 

 and the latter for angles of repose that are greater ; and that 

 the direction of the resulting force makes it in no small degree 

 a crushing force. 



It also appears from the equations, that when the angle of 

 repose is 45°, the face vertical, and top horizontal, that the 

 tangent of the angle of fracture is (l) equal half the tangent 

 of the angle of repose. The equation of Prony, for the same 

 case, gives the tangent of the angle of fracture equal to the 

 tangent of half the angle of repose. 



In the following Tableof Coefficients, for finding the maxi- 

 mum values of the resistances. 



Column 1 contains the engineering names for the slopes 

 corresponding to some of the angles of repose in Column 2. 



Column 2 contains the angles of repose from which the co- 

 efficients of t^;Ai^ are calculated. 



Column 3 contains the complements of the angles of re- 

 pose in column 2 ; or the angle which the direction of the re- 

 sulting resistance makes vihh the face, taking friction thereof, 

 into account. 



Column 4 contains the coefficients which, multiplied by 



