210 OF THE PRESSURE OF FLUIDS ON DYKES AND EMBANKMENTS. 



consequently, if d and b be respectively substituted for x and y in 

 the preceding equation, we shall have 



d^/7b^S (175). 



This equation involves the conditions necessary for preventing the 

 dyke from turning about the point B, and if the equation be resolved 

 into an analogy, we shall have b : d : : \/s : ^/sf. 



COROL. From which we infer, that the section is in the form of a 

 rectilinear triangle, whose base is to the perpendicular height, as the 

 square root of the specific gravity of the fluid, is to the square root of 

 the specific gravity of the wall or dyke. 



230. EXAMPLE. The perpendicular altitude of an embankment of 

 earth is 20 feet ; what must be the breadth of its base, so that each 

 portion of it estimated from the vertex, shall resist the effort of the 

 fluid, to turn it round the remote extremity of the base, with equal 

 intensity ; the water and the dyke having equal altitudes, and their 

 specific gravities being 1 and 1 .5 respectively ? 



Here we have given rfrz 20 feet, szr 1, and s r zz 1 .5 ; consequently, 

 by the preceding analogy, we have ^ 1.5 : ^ 1 : : 20 : 16.33 feet. 



231. The conditions necessary for preventing the portion of the 

 section ABG from sliding on its base, may be thus determined. 



We have seen (art. 220), that the momentum of the horizontal 

 pressure, to urge the section along its base, is m Jse?% 



consequently, by substituting x 9 for d 2 , we have m Jsa: 2 , 



but the momentum of the section opposed to this, is m' zz ns'fyx; 



therefore in the case of an equilibrium, we have isx* = ns'fyx, 



from which, by taking the fluxion, we obtain sxx~ ns'yx, 

 and by casting out the common factor, we get sx^nns'y. (176). 

 From this equation, when converted into an analogy, we shall obtain 

 x : y : : ntf : s. 



Which also indicates a rectilinear triangle, whose altitude is to the 

 base, as n times the specific gravity of the embankment, is to the 

 specific gravity of the fluid. 



If the water presses against the perpendicular side of the wall, the 

 curve bounding the other side, so that the strength of the wall may 

 be every where proportional to the pressure which it sustains, must be 

 a semi-cubical parabola, whose vertex is at the surface of the fluid, 

 and convex towards the pressure. 



