92 
Proceedings of Royal Society of Edinburgh, [jan. 31 , 
limiting amount when failure occurs along one or both of the two 
possible planes of rupture which may be shown to exist. 
The author ventures to advance the view that the condition of 
things may be rendered intelligible by a second application of 
Moseley’s principle. Tor, consider the stability of the wall, and 
suppose the critical condition to have been attained, and that so 
before /3 had reached the value /i 2 . To the actual value of J3 
looked upon for the moment as a limiting angle, there corresponds a 
certain roughness of the wall ; and suppose that by some external 
agency (such as a change of temperature or the like) the degree of 
roughness of the actual wall to be, in effect, reduced to this value. 
It is impossible to conceive that by a process such as this the 
critical state can have been disturbed. On the other hand, there 
can be no difficulty in perceiving that, if it is a fact that friction 
between the earth and the wall adds to its stability, failure may 
be brought about by simply allowing the degree of roughness to fall 
ever so little below the critical value — the value, that is, which 
corresponds to the actual value of /3 when the wall is in the critical 
state. Hence /3 has a value such that the stability of the wall in 
the critical state is a maximum ; or, in other words, the overturning 
moment of the earth pressure for any given wall is a minimum with 
respect to /?. 
In most cases which occur in practice, /3 may range through all 
values up to </> ; for not only are the inner faces of retaining walls 
usually left rough, but they are frequently stepped in a manner 
which makes a cross section somewhat resemble a flight of steps. 
The author has calculated some numerical values for s, corre- 
sponding to certain values of <£ and j3, which will be found in an 
annexed table. He has also tabulated the corresponding values of 
the ratio of breadth to depth of a wall whose density is equal to the 
density of the earth, and whose moment of stability is just equal to 
the overturning moment of the earth pressure. A polar curve 
showing the values of these ratios for <j> = 40° corresponding to the 
successive values of /3 is also annexed. 
Equation (14) resulting from (13) by putting /3 = 0, is identical 
with the corresponding formula of Rankine. The author has, how- 
ever, farther verified the formula (13) and by consequence (6), for 
the case where <£ = 60° and /3 = 30°. Inserting these values in (13), 
