ME. MACQUOEN EANKINE ON THE STABILITY OF LOOSE EAETII. 
19 
Corollary to the Theorems III., IV., and V. — When the angle of obliquity 6 is equal 
to the angle of repose (p, the quantities given by the equations (35.), (30.), (37.), have 
each but one value, without any limits of deviation, viz. — 
— 2’ L 
M=:]Isec(p; D = Iltan(p. 
(38.) 
§ 11. Planes of Bupture. 
The angle 'T', above given, is that made on either side of the axis of greatest pressure 
by the normals to the pair of planes along which the tendency of the earth to give way by 
sliding is greatest. The angle made by these planes themselves on either side of the axis 
of greatest pressure is therefore 
( 3 «') 
Those planes are called Planes of Bupture. Their position, in the particular case of a 
horizontal bank, where the axis of greatest pressure is vertical, was determined by 
C'ouLOMB by the aid of his ideal wedge of least resistance. 
§ 12. Application q/'Mr. Moseley’s Principle of LeoM Besistance to the Stahility of 
Earth. 
This principle may be stated as follows : — The forces which balance each other in or 
upon a given body or structiwe being distinguished into two systems, called respectively 
active and passive^ which stand to each other in the relation of cause and effect, then will 
the passive forces be the least which are capable of balancing the active forces, consistently 
with the physical condition of the body or structure. 
In a mass of earth, the active forces are the vertical pressures produced by the gravita- 
tion of its parts ; the passive forces are the pressures conjugate to those vertical pres- 
sures, whereby the earth is prevented from spreading. The pressures conjugate to the 
vertical pressm-es will therefore be the least which are at once consistent with the con- 
ditions of internal equilibrium given in §§ 6, 7, 8, and 9, and with the conditions of 
stability at each point, given in § 10. 
§ 13. Statement of the General Problem of the Stability of a Mass of Earth under its 
own Weight. 
The upper and free surface of the mass of earth, at which the intrinsic vertical pres- 
sure Xh = 0, is supposed to be curved in one coordinate plane only, that of greatest and 
least pressures ; and the form of the section of that sm’face by the vertical plane of 
greatest and least pressures is supposed to be given by an equation of the form (28.). 
The specific gravity G and angle of repose p of the earth being given, it is required 
D 2 
