1887.] Mr A. C. Elliott on Formula for Retaining Wall. 89 
Consider the forces acting on the prism : — 
(1) Its weight : this may he neglected in comparison with the 
other impressed forces, which are small quantities of the 
second order, while the weight is a small quantity of 
the third order. 
(2) The forces acting on the end faces of the right prism : — these 
are independently balanced as regards the wall, and may 
henceforth be left out of consideration. 
(3) The remaining impressed forces acting parallel to the plane of 
the section : of these let p v be the vertical pressure in the 
neighbourhood of the prism, due to the column of earth ; 
p 0 the inclined pressure exerted by the wall. 
/3 the angle between the direction of p 0 and the normal to the 
inner face of the wall, drawn outwards. This angle /?, in 
the case where motion is just about to take place along the 
interface BC, will be equal to the angle of friction for the 
earth on the wall. 
r the pressure of the contiguous earth on the face AB of the prism; 
a the obliquity of r ; or the angle which the direction of the stress 
r makes with the normal to AB ; 
0 the angle BAC. 
Considering first the equation of moments, it will appear that the 
tangential stress on AC must be equal to the tangential stress on 
BC ; but the tangential stress on BC is p 0 sin j3 , which is therefore 
the value of the tangential stress on AC. 
Besolving vertically and horizontally 
p v b+Posin/3. a~r cos (6 -a) . c = 0 . . . . (1) 
p 0 cos /3 . a+p 0 sin /3 . b - r sin (6 - a) . c — 0 ... (2) 
Dividing out by c and substituting in terms of 0, 
p v cos 6 +p 0 sin sin 0 = r cos (0 - a) .... (3) 
p 0 cos J3 sin 0 +p 0 sin (3 cos 6 = r sin (0 - a) .... (4) 
Eliminating r, and writing s for the ratio p 0 !p v 
1 sin /3 tan 6 1 + tan a tan 0 
s (cos j3 + sin j3 cot 6) 1 - tan a cot 0 
O) 
If now 6 be regarded as the independent, and a as the dependent 
variable, to find the condition for a a maximum (5) must be 
