18 
ME. MACQIJOEN EAXKINE ON THE STABILITY OF LOOSE EARTH. 
greatest obliquity of the pressure at any point in any plane traversing that mass, it appears 
from equation (2.) that the condition of stability of the mass ls 
e<<p (32.) 
From this condition the following propositions are deduced : — 
Tiieoeem I. At each 'point in a mass of earth, the ratio of the difference of the greatest 
and least pressures to their sum cannot exceed the sine of the aiagle of repose. 
This theorem* follows from the second of the equations (T.), its symbohcal expression 
being 
P,-P^_D . 
P,+ P 
(33.) 
Tiieoeem II. The follo'wing is the expression of the condition of staMlitij of a mass of 
earth, in terms of the pressures at a point, referred to any pair of rectangular oj:es. Ox'. 
Oy', in tdie plane of greatest and least pressures : — 
(P,,-Py)" + 4Q'"_D^_ 
<sin^ip (oA.) 
This follows from the equations (10.). 
Tiieoeem III. The following is the expression of the condition of the stability of a mass 
of earth, in terms of the ratio of a pair of conjugate pressures in the plane of greatest and 
least pressures : — 
Let E„, K„ be the two conjugate pressures, 6 their common angle of obhquit\"; then 
R„ > cos fi + i/ cos^ 9 — cos^ t ^ ) 
Rtt cos $+ \/ cos^ S — cos^ (p 
This follows from equation (15.). 
Theoeem IV. The positions of a pair of conjugate ptlanes being given, the following are 
the limits, consistent with stability, of the angle which the axis of greatest pressure can 
make with either of the normals to the conjugate planes : — 
sin 0 
sin p 
(36.) 
This folloAvs from the second of the equations (11. )• 
Tiieoeem V. The amount R and obliquity 6 of the qyressure on a given plane being 
given, the following are the limits, consistent with stability, of the half sum INI and half 
difference D, of the greatest and least principal pressures \ — 
M 
D=M sin & 
< 
R 
cosfl+ v/cos^fi— COS^(p 
R sin p 
cos $+ V cos^ 6 — cos'^ p 
(37.) 
This follows from the first of the equations (11.). 
The greater values of M and D given by this equation correspond to the greater values 
of the angle -vj/ given by equation (36.). 
* Already published in the ‘ Proceedings of the Royal Society ’ for the 6th of March, 1856. 
