OF THE PRESSURE OF FLUIDS ON DYKES AND EMBANKMENTS. 201 



COROL. The breadth at the base, as determined by this and the 

 preceding equation, exhibits but a small difference, being in excess in 

 the former case, by a quantity equal to 0.109 of a foot; but the 

 breadth at the top in the latter case, exceeds that in the former, by a 

 quantity equal to 1.891 feet; and the difference in the area of the 

 section, is 17.82 feet: it is consequently more expensive to erect a 

 dyke or embankment, with the side next the fluid perpendicular, than 

 it is to erect one of equal stability with both sides inclined or sloping 

 outwards. 



216. If the slope e should vanish; that is, if the side of the dyke, 

 opposite to that on which the fluid presses, be 

 perpendicular to the horizon, as represented by 

 A BCD in the annexed diagram, then, the equa- 

 tion (149) becomes 

 sd*=d$s(3b )-}-3Ds'(6 ? c)4-DcY. (157). 



But when the effect of the vertical pressure of 

 the fluid is omitted, we obtain 



sd s =i SDS' (b* c&) -f- DC 9 s', (158). 



and by supposing the altitude of the dyke, and the depth of the fluid 

 to be equal (the specific gravity of the fluid being expressed by unity) ; 

 then we have d D, and the foregoing equation becomes 



d* = 3s'(b* c&)4-cV; 

 consequently, by transposition and division, we get 



A* I * CV 



~~37~ (159). 



from which equation, the value of b is easily determined. 

 Let the slope of that side of the dyke on which the fluid presses, be 

 equal to 2 feet, and the perpendicular altitude of the dyke, or the 

 depth of the fluid 20 feet, the specific gravity of the material being 

 If as before ; then by substitution, the foregoing equation becomes 



,-==$!!=,, 



by completing the square, we obtain 

 p 2b + 1 = 74.8571 4- 175.8571 ; 



consequently, by extracting the square root and transposing, we get 



b = 9.709 feet. 



In this case, the dyke has less stability than it has when the perpen- 

 dicular side is towards the water, as is manifest from its requiring a 

 greater section, and consequently, a greater quantity of materials to 

 resist the effort of the pressure which tends to overturn it. 



