16 
ME. MACQUOEN SAJS’KIKE 0^’ THE STAEILITT OF LOO.SE EAETH. 
§ 8. Surfaces of Uniform Thrust and Unifoi'm Vei'tical Pressure. 
With an exception to be described in the next section, the only case in wbicb the 
equation (25.) becomes bnear with respect to x, and capable of being satisfied by an 
indefinite number of arbitrary forms of surface, is that in which each sm-face of uniform 
thrust is also a surface of uniform vertical pressure ; that is to say, when 
X=F(H) •. . . . (26.) 
In this case, the integral of equation (25.), as found by the method of Foueiee, is 
capable of being expressed in various forms, of which the following is the most compre- 
hensive : — 
(27.) 
the function f being such, that neither 
e~"'f nor e~'^f\ 
shall become infinite for any value of the argument, how great soever, nor increase 
indefinitely with the argument, and that they shall both vanish at the limits of integra- 
tion. This function is determined by the following condition at the upper surface of 
the mass : — 
^o=/(^) (28.) 
In all those cases in which the upper surface of the mass derfates alternately above 
and below an inclined or horizontal plane by deviations which recur periodically in each 
horizontal distance 2B, the integral of the differential equation (25.) is capable of being 
expressed in the following form : — 
*=^+Ajr+2.«-w(c.sm!g'+C.cos’f^), .... (29.) 
where A is the tangent of the average declivity, above and below which the smTaces of 
equal thrust deviate periodically, and C„ and are determined by the following for- 
mulae : — 
((To— A?/) sin ^ 
B -D 
2 
((To— Ay)cos^^ 
B JJ 
.dy 
.dij, 
► 
(30.) 
Such are the integrals of the equations of internal equilibrium in two dimensions in a 
vertical plane, in those cases in which each siu’face of uniform thrust is also a siu’face of 
uniform vertical pressm’e; a condition realized in those cases in which the horizontal thrust 
is caused by the vertical pressure. 
The relation X=F(H) between the horizontal thrust and the vertical pressure, still 
remains to be determined by the physical conditions of each particular problem. 
