ME. MACQrOEN EAJSTKINE ON THE STABILITY OE LOOSE EAETH. 
11 
Then the magnitude and dmection of the pressure exerted at the plane normal to On, 
are given by the following equations : — 
D sin 2^1/ 
(5.) 
tan^_j^_^j3^^g 2vl/’ 
or otherwise by the following : — 
P„=M+Dcos2^|/| 
Q„=Dsin 24 '. J 
The maximum value 0 of the obliquity 6, and the corresponding position of the 
normal On, are given by the following equations : — 
,D 
■^"= 7+0 8111 - 
M 
r^ • -1 D 
0= sm L , 
M 
( 7 .) 
to which correspond the following pressm^es, total, normal, and tangential ; 
P(^)=M(l-g) ^ (g) 
Q(’F)=D^iI; 
5!. 
Tig- 1- 
^ \ 13? 
Tt" 
The following geometrical construction represents 
the theorems expressed symbolically by the equations 
(5.), (6.), (7.), (8.) (fig. 1). Let xOx, yOy be the axes of 
gi’eatest and least pressure at the point O. Given, the 
half-sum M, and half-difference D, of those pressures, 
it is required to find : — 
Fii'st, the direction and magnitude of the pressure at 
a plane normal to On, which line is in the plane xy, 
and makes with the axis of greatest pressure the given 
angle xOn='^. Make 0^^=M. On Ox take the point a so that na=On. On na take 
nr=T). Then will Or=R„ represent the pressure requu’ed, and rOn=6 will be its angle 
of obhquity. 
Also, let fall rj)^On; then Op=:P„, j9r=Q„. 
Secondly, to find the plane for which the obliquity & is greatest. Make ^ON=‘T'= 
half a right angle -f- half the angle whose sine is Then ON will be the normal to a 
plane having the required property. There is obviously a pair of such planes, whose 
normals make equal angles at either side of Ox. 
The remainder of the construction is to be proceeded with as before, to find the total 
c 2 
