14 
ME. MACQrOEN EA^s^KIXE ON THE STABILITY OE LOOSE EAETH. 
of which the value, in terms of M and D, has akeady been given. Then it appears that 
R(^) 1 
M =- ^ 
cos© 
D =M sin0=E.(T)tan0 
2T^=^+0, 
(14.) 
in accordance with the equations (7.) and (8.). 
From the equations (13.) and (14.) it is easily deduced that the ratio of a pair of con- 
jugate pressm’es has the following value : — 
Rti cos — V sin"^ © — sin^ fl 
cos 3+ 'v/sin^© — sin^3 
(15.) 
§ 6. Lemmata as to the Intmial Equilibrium of a Solid Mass. 
Let 0.r', 0_^', Oz' be rectangular axes, of which Osd is vertical, and positive downwai’ds ; 
and let G be the weight of unity of volume of a solid mass. Then the well-known con- 
ditions of the internal equilibrium of such a mass are the following : — 
x' , ^ dOLy! p, 
dx^ dy' ' dz' ’ 
d I d^ yf , d ^ 
dx’ ' dij dz’ ’ ^ 
dGiyf ^ dOtx' I dP ^ 
dx' ' dy' ' dz' 
(16.) 
In all actual problems respecting the stability of earth, the plane of greatest and least 
pressures is vertical, and there is one horizontal direction, normal to that plane, along 
which the state of stress of the earth does not vary. It will be sufficient, therefore, to 
restrict the above equations to two dimensions, by making Q_,j=0 ; Qy= dz^—^ ; and 
putting Q simply for Q^,. Then we have the two equations, — 
dP,, 
dx' 
dx' 
dPy 
dy' 
: 0 . 
(17.) 
§ 7. Surfaces of Uniform Horizontal Thrust. 
The following is a peculiar transformation of these 
differential equations, suited to the subject of the 
present investigation. OX, fig. 2, being vertical, 
and OY horizontal, and in the plane of greatest and 
least pressure, conceive the mass to be subdivided 
into prismatic molecules by an indefinite number of 
vertical planes perpendicular to the plane XY, such 
as 1 ^ 2, 15 2 ^ 2, 25 and by an indefinite number of 
surfaces, also perpendicular to the plane XY, such 
as a,5i, ^ 2^25 and of such a figure, that the tangent 
