20 
ME. MACQUOEN EAXKIXE OX THE STABILITY OE LOOSE EAETH. 
to determine the form and position of the suidace at which the intiinsic yertical pressure 
has any given value. 
In order that the equations (27.), or (29.) and (30.) may furnish the complete solution 
of this problem, it is necessary now to determine, from the conditions of stability in § 10, 
the relation Xjj=F(H), between the intriusic vertical pressure and the horizontal thiiist. 
The case in which the upper smTace of the earth is an indefinitely extended plane, 
horizontal or inclined, is the only case which admits of an exact solution. It will there- 
fore be solved first, and its solution used to facfiitate the solution of the more complex 
case, which is solved approximately by the integral (27.), or by infinite series of the 
form (29.). 
§ 14. Pkoblem I. Surfaces of Equal Pressure and Thrust in the case of a horizontal or 
tmiformly sloqnng bank. 
In this case, equatiorr (28.), giviirg the form of the fiee smiace, becomes 
x^=Ay=ytdcn.6 (“iO-) 
Equatiorr (27.) gives for the form and position of any smiace of uniform intrinsic 
vertical pressure, 
x=^-\-A.y, orx—x„=^ (41.) 
Hence the surfaces of equal intrinsic vertical pressrne are planes parallel to the free 
surface, and the vertical pressure is simply the weight of a colrmrn of earth of irnity of 
area of base, aird of the height x—Xq. At each poiirt, a vertical plane, and a plane 
parallel to the free surface, are conjugate to each other ; that is, the pressure orr a plane 
parallel to the surface is vertical, and the pressure on a vertical plane is parallel to the 
surface ; and the angle of slope, 0, is the common angle of obhquity of those conjugate 
pressures. 
Equation (20.) gives for the vertical pressure per unit of area of a plane parallel to the 
surface, 
K„=Xcos^=G(^ — ^o)cos^ ("i-O 
Now from the principle of least resistance, it follows that the pressm-e at any pomt on 
a vertical plane, in a direction parallel to the slope, must have the least value consistent 
with the equation of stability (35.); that is to say. 
cosfl— V^COS^ fi— cos® (p 
“ cos + v” cos® — cos® (p 
Eor brevity’s sake, let 
coo 2 ^. cps^— Gcos®6— cos®ip _^. 
cos $+ V cos® 6 — cos® f 
(43.) 
(44.) 
Then the horizontal component of has the following Aulue : — 
/7H 
^ cos 0=kX.=zkG{x-Xo). . 
(46.) 
