94 Proceedings of Royal Society of Edinburgh. [jan. 31 , 
The force on an element of area 1 foot long horizontally, and of 
depth dh is therefore 
dV = shp e dh. 
Hence the whole action of the earth on the wall per foot-run is 
represented by a force 
P=2 fyi s P •••••••• (16) 
where b 1 is the effective depth of the earth. P will act at the 
centre of pressure, and be inclined at the angle /3 to the horizontal. 
Let b be the breadth, h the height, and p m the density of the wall. 
Then, equating the overturning moment to the moment of stability, 
and solving a quadratic 
b Pe $ ssm/3 f is sin /?\ 2 pm cos/? ) 
ftVsa ~ ^" + vnr-j + h $ ~s~r • • (1,) 
A good example of the divergence of the results obtained by 
Rankine’s formula from actual facts, even where the earth mass 
approximates to the hypothetical granular material of the mathe- 
matical investigation, is adduced by Mr Baker in his paper, “ On 
the Actual Pressure of Earthwork,” already referred to. The 
example in question is not only interesting in itself, but has the 
additional advantage of having been selected by Mr Elamant for an 
application of Professor Boussinesq’s formulae. 
Mr Baker says — “When the wood paving was recently laid in 
Regent Street, the space being limited, the stacked wooden blocks 
in many cases had to do duty as retaining walls to hold up the 
broken stone ballast required for the concrete substructure. In one 
instance (Example I.) the author noted that a wall of pitch pine 
blocks, 4 feet high and 1 foot thick, sustained the vertical face of a 
bank of old macadam materials which had been broken up, screened, 
and tossed against this wall, until the bank had attained a height 
of 3 feet 9 inches, a width at the top of about 5 feet, and slopes 
on the farther sides, deviating little from P2 to 1” (Proc. Inst. 
C.JE., vol. Ixv. p. 145). [P2 to 1 corresponds to an angle of 
repose, 39° 48']. 
ISTow, Rankine’s hypothesis amounts to putting /? = 0 in the 
notation of the present paper. Putting therefore in (17) 
