498 Prof. Sylvester on the Pressure of Earth 



If, as in the case with which we shall have to deal, the sole 

 impressed force is that of gravity, and if we treat the weight of a 

 unit of the mass as unity, and make the axis of x horizontal and 

 of y vertical, the equations become 



dh ,£M 



dy dx ' 



dx dy ' 



These, being the equations which control the law of the variation 

 of the thrusts estimated in given directions in passing from one 

 stress to another, I call the equations of variation of stress. 



I now proceed to the application of the principles above set 

 forth to the treatment of the particular question in hand. 



Let fi be the coefficient of friction of the. earth upon itself, 

 and /i=tan X, so that X is the angle of repose; by this is to be 

 understood that the thrust on any element can never make, 

 with the perpendicular to that element, an angle greater than 

 X. Now the general law of the distribution of stress proves 

 that the actual angle between the perpendicular to the element 

 and its thrust will in two directions be zero. Hence at any 

 given point it will pass through all gradations, from zero up to a 

 certain limit. Here presents itself the question, Is that limit X, 

 or can it be X for every point in the mass ? As we have no right 

 to assume a priori that this limiting angle in that state of equi- 

 librium which we wish to determine must be equal to X through- 

 out the mass, and obviously it will not be so for actual cases of 

 equilibrium which arise, we want a name to distinguish the 

 maximum ratio which friction bears to pressure in any specified 

 stress from the absolute maximum which this ratio is capable of 

 attaining. We may name the former the coefficient of frictional 

 energy ; and for every point where this is equal to the absolute 

 coefficient of friction, we may say the friction of the stress is at 

 its maximum energy. Let (fi) be the coefficient of frictional 

 energy for any given stress, and (X) =tan -1 (/*) the corresponding 

 angle of repose. [We may also, if we please, term (//,) and (X) 

 the relative coefficient and relative angle of repose respectively, 

 t. e. relative to any assigned stress.] Let the ratio between the 

 maximum and minimum thrust of any stress be called 7*: a 

 simple relation connects 7 and (X) *. 



For calling, as before, L the pressure, and M the face-force 

 (now the friction), we have by equation (1), 



* This relation and its importance are well known to Professor Rankine. 



