1887.] Mr A. C. Elliott on Formula for Retaining Wall. 91 
cos (3(\+2 tan 2 <f>) ± 2 sec <j> J cos 2 /? tan 2 </> - sin 2 (3 . /jq\ 
4 sin 2 j3 sec 2 </> + cos 2 (3 
giving the relation between s, /3 , and </>, desired. 
(10) may be written in a form more convenient for calculation 
thus — 
cos /3{2 — cos 2 </>] ± 2 */cos 2 /? — cos 2 </> on 
4 - cos 2 /?{4 - cos 2 </>} 
Again, when (3- 0 (11) may be written 
s= l±sin^, ....... (12) 
1 + sin </> 
And it will also be observed that, when (3 = </>, the quantity under 
the radical sign becomes zero, and when /?></>, negative,- rendering 
the whole expression imaginary from a physical point of view. 
These equations give two values of s consistent with the condi- 
tions. Since, however, the reaction of the wall is a 'passive force, 
corresponding to the active pressure of the earth, it appears at once 
from Moseley’s principle that the smaller value of s must be taken. 
Therefore, finally, 
_ cos (3 [2 — cos 2 </>} - 2 Jcos 2 (3 — cos 2 </> 
4 — cos 2 /? {4 - cos 2 </>} 
and 
1 - sin </> 
s=- r — v when B 
1 + Sill <f> 1 
0 
(13) 
(14) 
Given therefore the angle of repose of the earth, and the correspond- 
ing friction angle for the earth on the wall, the direction and amount 
of the mutual action becomes determinate if we assume that failure 
cannot take place until (3 has attained its limiting value, which may 
be denoted by (3 9 . It may be at once remarked that, however 
closely (3 may approach to </>, it can never exceed that value ; since, 
when /? = </>, the surface of separation, between the wall and the 
earth, becomes a plane of rupture. When, on the other hand, 
it is supposed with greater generality that at the time of fracture 
/? 2 >/?<</>, equilibrium must be destroyed without any relative 
motion between the wall and the earth in the immediate neighbour- 
hood of the inner face having, in the first instance, taken place. In 
other words, the inclination denoted by j3 has not reached its 
