IVIE. MACQUOEN EANKINE ON THE STABILITY OE LOOSE EAETH. 
13 
To adapt this theorem to the present question, the first step is to transform the equa- 
tions (5.), so as to make the obliquity of the pressm-e, 0, the given angle, instead of the 
angle of direction -ip of the normal to the plane. Thus are obtained the following equa- 
tions, from which, when the greatest and least pressures at a point are given, there may 
be found the position of a plane perpendicular to the plane of greatest and least pressure, 
on which the obliquity of the pressure shall be equal to a given angle ^ ; and also the 
amount K„ of the pressure corresponding to such obliquity. 
=M cos — M^sin^^ 
or ’^1/11 .jMsinS 
2-<P=-^+^±cos — 
( 11 .) 
Hence it appeal^, that for each value of the obliquity there are two values of \p and 
two of E, the less value of -p/ corresponding to the greater value of E, and conversely. 
Let -pa be the less, and p„ the greater value of p, E„ the greater, and E^ the less value 
of E. Let the two normals Ou, Ov be drawn at opposite sides of the axis of greatest 
pressure 0.t ; then the angle between them is 
uOv=p,-{-P,=^-i-^; 
and the angle between the planes to which they are normal is ■ 
'^ — pu — pv = 2~^'’ 
(12.) 
therefore those two planes are conjugate. 
Proble^n . — The positions of a pair of conjugate planes, both perpendicular to the plane 
of greatest and least pressure, and the pressm'es on them, being given, it is required to 
tind the position of the axes of greatest and least pressure, and the magnitude of the 
greatest and least pressures. 
From the equations (11.) it is easily deduced, that 
M 
Rk + Rp 
2 cos 6 
D =M\/ (1^; cos o)‘} 
— X' + E„)^} 
2-4'=^+^ + COS *— TT 
\/V + 
E„— Rp 
R« + Rp 
cotan $ 
(13.) 
The axis of greatest pressure will be found in the obtuse angle between the normals Ou, 
Ov, and nearer to Ou, the normal to the plane on which the pressure is the greater. 
A^Tien E„=Ep, then ^=0, the angle of greatest obliquity. In this case let 
E„=EpStEC^), 
