12 
m. MACQL'OEN EANiaNE ON THE STABILITY OF LOOSE EAETH. 
pressure OK, and its normal and tangential components OP, PE. It is obyieus that 
OKN is a right angle. 
The locus of the points r is an ellipse, whose semiaxes Oj:=P,, represent the 
greatest and least pressures at the point O. 
§ 4. Additional Lemmata, as to the Transformation of Stress in the plane of greatest and 
least pressure. 
By the plane of greatest and least pressure at a given pomt O, is meant the plane con- 
taining the axes Ox and Oy of greatest and least pressure. Let there be taken any two 
new rectangular axes in that plane. Ox' and Oy', and let 
~c:xOx'=<yOy'='ip, so that <xOy'=^-\--4>. 
Also let P^, and Ty, be the normal pressures at planes perpendicular to the axes OP, Of, 
respectively, and Q' the tangential pressure on either of those planes. Then from equa- 
tions (6.) it appears that 
P^,=M-1-D cos 2-ip' 
Py=M — D cos 2-p > 
Q' =D sin 2-p. 
(9.) 
Consequently, if the elementary stresses P^,, Yy,, Q', at any pair of planes at right angles 
to each other and to the plane of greatest and least pressure be given, the greatest and 
least pressures, and the positions of their axes, are given by the following equations : — - 
P +P P - 
1A_J' = M = — 
p -p 
-y y 
. o , 2Q' 
tan 2p — p - 
A td 
( 10 .) 
The equations given above solve a particular case only of the general problem, \iz. 
the case in which the given elementary stresses act in the plane of greatest and least 
pressm’e. But in all actual problems respecting the stability of earth, the plane of greatest 
and least stress is known ; and it is therefore unnecessary to apply to that subject the 
general problem as to the finding of the axes of pressure in space of three dimensions ; a 
problem which requires the solution of a cubic equation. 
^ 5. Lemmata as to Conjugate Planes and Pressures. 
It is a well-known theorem in the theory of the elasticity of sohds, that if the pressure 
on a given plane at a given point be parallel to a second plane, the pressm-e on the second 
plane at the same point must be parallel to the first plane. Such planes ai’e said to be 
conjugate to each other, with respect to the pressures on them ; the pressures also are 
said to be conjugate. 
