IIR. MACQUOEN EANKINE ON THE STABILITY OF LOOSE EAETH. 
21 
And the total horizontal thrust, from the sm'face down to a given depth is 
kG{x—Xo)^ 
2 (46.) 
The relation, then, between the vertical pressm'e and the total horizontal thrust down 
to a given sm’face of uniform thrust and pressure, is expressed by the equation 
kX^ ^ /2 
GH 
k 
(47.) 
Equation (36.) gives, for the angle made by the axis of greatest pressure at each point 
with the vertical. 
TT , TT 1 , sin fl 
V= 7 ~o— 77COS — 
2 “ 4 2 2 sin <2 
(48.) 
This axis hes in the acute angle between the slope of the surface and the vertical. 
The two planes of rupture at each point make -with this axis the angle given by 
equation (39.) ; that is to say, they make with the vertical the follo’wing angles respectively 
at opposite sides: — 
TT 
2 
fl 1 
2 + 2 COS 
sin 9 p 
sin (p 2’ 
sin S 
sin (p 2 
(49.) 
§ 15. Extreme cases of this Problem. 
The two extreme cases of this problem are respectively, when the free surface is 
horizontal, or 6=0, and when it slopes at the angle of repose, or 6=(p. 
The following are the results in these two cases. 
Case First. ^=0: 
Il„=X; E„=X 
1 — sin ip 
1 + sin <p ’ 
1 — sin (p . 
1 + sin ip ’ 
X 
2 
— \p = 0, or the axis of greatest pressure is vertical, and the planes 
>. . . . (50.) 
of 
ruptm’e make the angles ^ f on either side of the vertical. 
Case Second. 0=(p: 
E„=Xcos®; K„ = R„ = Xcos(p; k=cos^(p; 
T . X (p . 
Q — v = 2 ? or the axis 01 greatest pressure bisects the acute angle ^ 
between the vertical and the slope of the surface, and the planes of 
ruptme are respectively vertical, and parallel to the smface. 
(51.) 
§ 16. Bevetement Wails. Peoblem 11. To determine the magnitude and direction of 
the resultant pressure on the base of the revetement wall of an earthen bank, having a 
horizontal or uniformly sloping upper surface, the angle of repose of the earth upon the 
masonry being not less than that of the earth upon itself. 
Case 1. When the back of the wall does not overhang the base. (See fig. 3.) 
