206 OF THE PRESSURE OF FLUIDS ON DYKES AND EMBANKMENTS. 



Suppose therefore, that the resistances of adhesion and friction, are 

 equal to n times the weight of the dyke, which we have represented 

 by w ; then we have 



nw \d*s\ 

 but we have shown, in the investigation of the preceding case, that 



w = jD/(26 c e}\ 

 consequently, by substitution, we obtain 



d 2 s = i>nJ(2b c e). (168). 



This is the equation of equilibrium, or that in which the resistance 

 of the dyke is counterpoised by the horizontal pressure of the fluid, 

 the effect of the vertical pressure not being considered ; but in order 

 to express the breadth of the base in terms of the other quantities, let 

 both sides of the equation be divided by ons', and it becomes 



D7ZS 



consequently, by transposition and division, we obtain 



d*s 



6 = 2^7+ i(c + e): (169). 



and finally, if the perpendicular depth of the fluid and the height of 

 the dyke are equal, we shall have 



- - . (170). 



225. In order therefore, to illustrate the reduction of the above 

 equation;1by means of a numerical example, we must assume a value 

 to the letter n, having some relation to the nature of the materials of 

 which the resisting obstacle is constructed ; now, it has been found 

 by numerous experiments, that when rough and uneven bodies rub 

 upon one another, or when a heavy body composed of hard and 

 rough materials, is urged along a horizontal plane, the effect of the 

 friction is equivalent to about one third of the weight of the body 

 moved ; or in other words, it requires about one third part of the force 

 applied to overcome the effects of the friction ; and moreover, in the 

 case of a wall built of masonry, there is, in addition to the friction, 

 the adhesion of the materials to the plane on which the wall is built. 



If therefore, we consider the effect of adhesion to be equivalent to 

 the effect of friction, it is manifest, that their conjoint effects will 

 destroy about two thirds of the force applied ; consequently, in the 

 case of masonry, we may suppose that the value of w, is very nearly 

 equal to 1J, but for other materials it will vary according to the 

 specific gravity or weight. 



