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II. On the Stability of Loose Earth. By W. J. Macquorn Rankine, F.R.S. 
Eeceived June 10, — Eead June 19, 1856. 
§ 1. General Prindjple. 
The subject of this paper is, — the mathematical theory of that kind of stability, which, 
in a mass composed of separate grains, arises wholly from the mutual friction of those 
grains, and not from any adhesion amongst them. 
Previous researches on this subject are based (so far as I am acquainted with them) 
on some mathematical artifice or assumption, such as Coulomb’s “ wedge of least resist- 
ance.” Researches so based, although leading to true solutions of many special problems, 
are both limited in the apphcation of theii’ results, and unsatisfactory in a scientific point 
of view. I propose, therefore, to investigate the mathematical theory of the frictional 
stabdity of a granular mass, without the aid of any artifice or assumption, and from the 
following sole 
Principle. 
The resistance to displacement by sliding along a given plane in a loose granular mass., is 
egual to the normal pressure exerted between the parts of the mass on either side of that 
plane., multiplied by a specific constant. 
The specific constant is the coefficient of friction of the mass, and is regarded as the 
tangent of an angle called the angle of repose. Let P denote the normal pressm-e per 
unit of area of the plane in question ; F the resistance to sliding (per unit of area also) ; 
(p the angle of repose ; then the symbohcal expression of the above principle is as 
follows : — 
F 
p=tan^ (I.) 
This principle forms the basis of every investigation of the stability of earth. The 
peculiarity of the present investigation consists in its deducing the laws of that stability 
from the above principle alone, without the aid of any other special principle. It will 
in some instances be necessary to refer to Mr. Moseley’s “ Principle of the Least Resist- 
ance ; ” but this must be regarded not a special principle, but as a general principle of 
statics. 
§ 2. Corollary as to Limit of Obliquity of Pressure. 
It is necessary to the stability of a granular mass, that the direction of the pressure 
between the portions into which it is divided by any plane should not at any point make 
with the normal to the plane an angle exceeding the angle of repose. 
MDCCCLVII. c 
