I^IE. MACQUOEN EAA^IQNE ON THE STABILITY OF LOOSE EAETH. 
case there is but one relation, consistent with stability, between the vertical pressure' and 
horizontal thrust, Avithout any limits of deviation. 
The relation betAveen those forces Avhich is the only solution in the case specified, is O'ue 
of the solutions consistent with stability in every other case, as will appear from tlie 
foUoAving 
Lemma. The intensity of the horizontal thrust, E„cos^=Xcos^ip, which corresponds 
with a given vertical pressure, X, for a declivity sloping at the angle of repose, (p, lies 
between the limits consistent with stability for every declivity sloping at a less angle. 
For the condition of stability deduced from equation (35.) is 
E, cos I 
E- 
-= cos 
•tvi 
2/1> 
cos^ 6 • 
COS^ (P 
cos^ S V ^ cos 5 j 
cos® 9 : 
cos® (P 
. . (54.) 
cos® 9 \ '' COS" 
and cos^ p is ahvays Avithin the limits fixed by this equation. 
§ 18. Pkoblem III. The vertical section of the upper surface of a mass of earth, 
curved in the plane of section only, being given, it is required to find the form and position, 
of the surfaces of uniform vertical pressure, when the greatest declivity of each of them is 
the angle of repose. 
It appears from the preceduig lemma that the solution of this problem is ahvays one of 
the solutions eA en AA'hen the greatest declhity is less than the angle of repose. 
Let the equation of the upper sm’facebe expressed by the equation (28.), or developed 
by the formulae (30.). Then the form of the siu’faces of equal pressure is gwen by the 
transcendental parts of the equations (27.) or (29.); and it only remains to determine 
their by finding the relation between the A'ertical pressure X and the horizontal 
thrust H. 
The condition of stability gives 
Ep cos 9 1 
cos® <p 
cos® ^ 
dx 
m 
(55.) 
From the general chfferential equation (25.), it appears that 
dx 1 ' d^xX . 
SH=G(rfH+^d> 
but at the points of greatest declivity of each smTace 
— -0 
consequently equation (55.) becomes 
X= 
G 
dX' 
which being integrated, gives 
H: 
cos^^.X® _ -v/2GH 
2G ’ 
being the solution required. 
cos <p 
(57.) 
