ME. MACQrOEX EAA^KIXE OX THE STABILITY OE LOOSE EAETH. 
Let ABDE be the section of the wall. Through A di’aw AT 
vertical (=^ — ^ 0)5 cutting the upper surface of the bank in F. 
If the back of the wall is vertical, F coincides until B. 
Through C, the centre of gravity of the mass of masomy and 
earth AFBDE, draw CW vertical. Take AH=^AF, and draw 
HR parallel to the smTace of the bank, cutting CW in O ; this 
will be the position and direction of the resultant pressm’e on the 
plane AF. To represent that resultant pressure, take 
ok=A="^ 
cos 5 2 cos 9 
Eig. 3. 
(52.) 
Also take OW to represent the weight of the mass of earth and masoniy AFBDE ; 
then will the resultant OS of OR and OW represent in magnitude and direction the 
resultant pressure required on the base AE. 
OS= A/{0R'+0W'+20R.0W.sin^}; 
sin<WOS=i; 
(63.) 
sin 6 is to be taken as positive when the bank slopes down tow'ards the ivall. 
Case 2. When the back of the wall overhangs, as represented, for example, by AT. 
Proceed in all respects as above, except that in finding the centre of gravity C, and 
the weight OW, the mass AFBDE alone is to be treated as masonry, and the prism AFT 
is to be treated as if its specific gravity were merely the excess of the specific gravity of 
the masonry above that of the earth. This is because a pressm’e equal to the weight of 
the earth wdiich the prism AFT could contain, is sustained by the earth vertically below 
it, leaving only the excess of the weight of masom-y over that of earth to add to the 
stability of the wall. As this excess is in general very small, it follows that there is in 
general little or no advantage in building revetement walls so as to overhang behind. 
For the same reason it is evident, that if a straight line be di’awm from A to U. where 
the line of slope cuts the face of the wall, the masonry behind this hue contributes little 
to the stability of the wall. 
17. General Case: — its amhigiiity. 
The application of the principle of least resistance to the case in which the vertical 
section of the upper surface of the earth is of any form (limited only by the condition 
that the slope shall nowhere exceed the angle of repose), is attended with this difficulty; 
— that at certain portions of each layer of equal thrust, \iz. the lower portion, it is 
difficult, if not impossible, to determine which of the two conjugate pressiu’es, the vertical 
pressm’e and the horizontal or inclined pressure, is to be regarded as cause and which 
•as effect ; and thus the solution becomes ambiguous. 
There is one case, however, which is not affected by such ambiguity ; 'viz. that in which 
the steepest declmty of each surface of equal thrust is the angle of repose ; for in that 
