OF THE PRESSURE OF FLUIDS ON DYKES AND EMBANKMENTS. 205 



Now, according to the property of the centre of gravity, HC is 

 equal to two thirds of BC, while h b is only one third of be ; but the 

 horizontal pressure of the water is the same in both cases ; it will 

 therefore require the same mechanical energy to resist it; and since, 

 by the conditions of the problem, the altitudes A B and a b are equal, 

 it follows, that in order to produce an equilibrium, the product of the 

 base of the triangle a be, into the length of the lever h b, must be 

 increased in such a manner, that 



BCXHC = 6cX^&, (166). 



and by converting this equation into an analogy, it becomes 

 BC : be : : hb : HC. 



We have seen, that by the construction and the property of the 

 centre of gravity, the lever HC is equal to two thirds of BC, and h b 

 equal to one third of be; let therefore, fsc and %bc, be substituted 

 for HC and h b, in the equation marked (166), and we shall obtain 



or dividing both terms by -^ , it becomes 



2BC 2 = C 2 , 



and finally, by extracting the square root, it is 



(167). 



Hence the reason for, and the nature of the increased breadth 

 become obvious, the one being the side, and the other the diagonal 

 of a square. 



Now, we have found that the breadth of the dyke at the base, is 

 equal to 10.68 feet, when the perpendicular side is in contact with 

 the fluid ; consequently, when the pressure is exerted on the hypo- 

 thenuse, we have 



6=10.68X1.4142 = 15.11 feet, 



being the very same result as that which we obtained from the reduc- 

 tion of the equation (165). 



224. What we have hitherto done, has reference to the case in 

 which the obstacle yields to the pressure of the fluid, by turning upon 

 the remote extremity of its base ; we have therefore, in the next place, 

 to investigate the conditions of equilibrium, when the obstacle is sup- 

 posed to yield, by sliding along the horizontal plane on which it is 

 erected. 



Since the base of the dyke or wall is horizontal, it is manifest that 

 the mass which it sustains, resists the horizontal pressure of the fluid, 

 only by its adhesion to the base, and the resistance occasioned by 

 friction. 



