ME. MACQUOEN EANKINE ON THE STABILITY OE LOOSE EAETH. 
15 
plane at each point of each of those surfaces is conjugate to a vertical plane, in the sense 
explained in section 5. Let 
a=Oa 
be the vertical ordinate of any one of those surfaces corresponding to ^=0 ; then 
(18.) 
will be its equation. By the definition of conjugate planes, the pressure on each element 
of such a surface is vertical. Let be its amount per unit of area of the surface, and E.^ 
the pressure on unity of area of a vertical plane, which pressm'e is parallel to a tangent 
to ah ; the angle of obliquity being given by the equation 
tan 6— 
dx 
dy 
(19.) 
Let X be the vertical pressure on a given element of a surface ah^ per unit of area of the 
prelection of that surface on a horizontal plane ; then 
X=A^=E,^i+' 
cos fl V , 
( 20 .) 
dx^ 
cos a ” V dy^ 
It is evident that the equations of the equilibrium of a prismatic element m are the 
following : — 
1 
d 
dy 
dx 
cos ^- 7 - ) =0. 
da 
( 21 .) 
The second of these equations being integrated, gives 
R — 
„ .dx ’’da -n/ \ 
R„ COS = j =F((2), 
da 
/ d^ 
V^-^df 
which value being introduced into the first equation, gives the following. 
dPx 
-(X-G^) + F(«).;f,= 0. 
dy^ 
( 22 .) 
(23.) 
Now let 
H= I 'E{a)da=\ 'R„c,o^ &-^da (24.) 
be the total horizontal thrust of the solid mass from its upper surface, down to the surface 
under consideration. This quantity, being independent of y, may be used as an inde- 
pendent variable instead of a; that is to say, dividing equation (23.) by F(a), we obtain 
the following : — 
■p^{Gx—X.)=y^, . (25.) 
which is the differential equation of a Surface of Uniform Thrust. 
