277 



If the top lying between F and D be loaded with a given 

 weight, the values of (p and R are rigorously determined from 

 the equations by producing the top ED to d, so that the tri- 

 angle CDd, multiplied by w anh the length of bank acted on, 

 may be equal to the given weight, and then substituting the 

 new values of h, S, 6, and j3, corresponding to the face Cd and 

 top Erf, in the equations, in place of those to the face CD and 

 top ED. 



When the resistance is generated by the pressure of the 

 bank against a structure at the face, S may be taken equal 

 2c'. In this case. 



tan j3 V(tan 9 tan 2c') 

 *^" ^ " V (tan d tan 2c') + V { (tan $ - tan /3) x (tan (i + tan 2c') ) ' ^ ^ 



ivli^ sin /3 tan j3 tan 6 1 



1^ = ^T~ o 



2 cos 2c 



( • ) [ ''' 



V v/ {(tan 2c'(tan0- tan/3 ) ) 1 + V {tan (tan 2c'+ tan (5)] J' j 

 When the face is vertical and the top horizontal, c = /3 : in 



this case 



cos c _^ 



wh^ sec c' / 1 



R = 



\ V* 2 tan c + sec c'J' ^ ^ 



2 



The value of tan (p here derived is equivalent to that of 



in equation (F) of Tredgold ;* but the value of the resistance 

 differs materially from his, and is far more simple. Tredgold's 

 equation (G) for the value of the resistance acting hori- 

 zontally, after making the necessary changes to our notation, 



is 



h^iv 1 



R = -^x 



2 . , ,„ , sin^ c' V 2 + sin^ c' V2 

 sin cv 2+1+ X—, — 



2 cos c 



Philosophical Magazine, vol. li. p. 402. 



