OF THE PRESSURE OF FLUIDS ON DYKES AND EMBANKMENTS. 209 



Take any point G in the line AGD, and through the point o thus 

 assumed, draw the horizontal ordinate GB, cutting the vertical axis 

 AC in the point B : now, it is required to determine the nature of the 

 curve AGD such, that each portion of the dyke, or of its section, as 

 AGE, estimated from the vertex, may be equally capable of resisting 

 the horizontal pressure of the fluid exerted against AG ; or which is 

 the same thing, that each portion may retain its stability and remain 

 in equilibrio on its base GB; not separating from the lower portion 

 GBCD, either by turning about the point c as a centre of motion, or 

 by sliding in a horizontal direction along the base GB. 



Put x AE, the abscissa of the curve estimated from the vertex 



at A, 

 y = EG, the horizontal ordinate corresponding to the ab- 



scissa x, 

 s z= the specific gravity of the fluid, which endeavours to 



displace the dyke by pushing it along the lin BG, 

 / zn the specific gravity of the materials of which the dyke 



is constituted, 



m m the momentum of the horizontal pressure, 

 m' the momentum of the resistance offered by the dyke, and 

 n :=z the number of times that the adhesion and friction of the 



dyke are equal to its weight. 



Then we have already seen, equation (145), that the momentum of 

 the horizontal pressure of the fluid as referred to the point c, is 



from which, by substituting x s instead of d 3 , we obtain m 

 which equation indicates the momentum of pressure at the point D. 



But the momentum of the resistance offered by the wall, that is, 

 the momentum of the portion of the section represented by ABG, is 



m' = is'fy*x; 



and these momenta in the case of an equilibrium must be equal to 

 one another ; hence we have 



from which, by taking the fluxion, we shall obtain 



or by suppressing the common factors, it becomes sx^ s'y*; 

 by extracting the square root of both terms, we get x\/s y\/s'. 

 Now, when x becomes equal to d, the whole perpendicular depth 

 of the fluid, or the altitude of the section ; then y becomes equal to 

 b, the thickness of the dyke, or the greatest breadth of the section ; 

 VOL. i. p 



