194 OF THE PRESSURE OF FLUIDS ON DYKES AND EMBANKMENTS. 



but by reason of the similar triangles EHG and EFD, we have the 

 following proportion, viz. 



d : 3 : : y : x, 



from which we obtain 



: 



and because the fluxions of equal quantities are equal, it is 



Let these values of x and x, be substituted instead of them in the 

 preceding value of m', and we shall obtain 



consequently, by taking the fluent, it becomes 



there being no correction, since the whole expression becomes equal to 

 nothing when y is equal to nothing. 



When y becomes equal to d the whole perpendicular height of the 

 fluid, then the foregoing value of m' becomes 



wf = 8d(J6 >8). (146). 



The foregoing equations (145) and (146), exhibit the horizontal 

 and vertical momenta of the pressure exerted by the fluid on the 

 sloping side of the obstacle ; and it is manifest from the nature of 

 their action, that they operate in opposition to one another ; the 

 horizontal pressure, endeavouring to turn the body round the point c 

 as a fulcrum or centre of motion, and the vertical pressure tending to 

 turn it the contrary way round the same point, or otherwise to render 

 it more stable and firm on its foundation. 



208. But the stability of the dyke is farther augmented by means 

 of its own weight, which being conceived to be collected into its 

 centre of gravity, opposes the horizontal pressure of the fluid with a 

 force, which is equivalent to its own weight drawn into a lever, whose 

 length is equal to the perpendicular distance between the centre of 

 motion, and a vertical line passing through the centre of gravity of 

 the section ABCD. 



Now, it is manifest from the principles of mensuration, that if the 

 transverse section of the dyke be uniform throughout, the weight is 

 proportional to the area of the section, multiplied into the specific 



