OF THE PRESSURE OF FLUIDS ON DYKES AND EMBANKMENTS. 193 



perpendicular depth of its centre of gravity, and again by the specific 

 gravity of the fluid ; it follows, that the horizontal pressure on the 

 increment of EG, is 



but by the principles of mechanics, the aggregate or accumulated force, 

 with which the horizontal pressure operates to overturn or remove the 

 dyke, is 



and by taking the fluent of this, it is 



/= %sy*. 



But the perpendicular distance from E, at which this force must be 

 applied, is manifestly equal to %y ; for the centre of gravity of the 

 triangle EHG, occurs in the horizontal line passing through that point; 

 therefore, the length of the lever on which the force operates to over- 

 turn the dyke is 



consequently, for the momentum of the force, we have 



and when y becomes equal to d, the whole height of the fluid, it is 



m sd\ (145). 



Again, the vertical pressure exerted by the fluid on the increment of 

 EG, is obviously equal to the weight of the incumbent column ; that is 



and this pressure expresses the force, with which the fluid operates 

 vertically to retain the obstacle in its position, or to prevent it from 

 rising to turn about the point c ; consequently, 



p'=f'=syx. 



Now, the length of the lever on which this force acts, is evidently 

 equal to ic, the distance between the fulcrum c, and the point i, where 

 the perpendicular passing through G cuts the base DC ; but ic accord- 

 ing to the figure, is equal to DC DF -f i F ; that is 



ic = b + *; 

 consequently, the momentum of the force /', is 



m' = syx(b 3 -far), 

 or taken collectively, the momentum on EG, is 



VOL. I. 



