278 



This value, however, is erroneous, and should be 

 h?w 1 



•tt = — t: 



2 . , ,„ , sin^ c' V 2 + ^ sin" c + sin c V2 



sin c V 2 + 1 + ^^ r^ '^— 



cos-c 



which, multiplied by sec c, to find the resulting resistance, is 



equal to the more simple form found above. 



When = /3 the top slopes upwards at the angle of repose : 



in this case 



tan (^ = tan /3, (7) 



wh? sin^ j3 

 ^ = ~2"~'' sin(2c'+|3)' ■ ^^^ 



The second of these equations gives the greatest of the maxi- 

 mum values of the resistance : if the face be vertical, tan /3 = 



. , and 



tan c 



R = — - cos c . (9) 



The horizontal portion of this resistance is 



K = — - cos- c = — - sin^ c. (10) 



As this value is the same as (7*) the limiting value of the 

 horizontal resistance, neglecting friction at the face, it appears 

 that the limiting value of the horizontal resistance is the 

 same whether friction at the face be taken in the calculation 

 or neglected. 



When the top slopes downwards at the natural slope, 



tan ^ = tan ij3, (11) 



„ _ wA^ sin /3 tan j3 / V (tan 2c' + tan j3 y 



2 cos 2c' Vtan 2c' sec /3 + tan 2c' + tan /3 j 



The value of the resistance here given is the least of the maxi- 

 mum values. If the face be vertical, 



* Proceedings, vol. iii. p. 86. 



