OF THE PRESSURE OF FLUIDS ON DYKES AND EMBANKMENTS. 195 



gravity of the material of which it is composed, and again into ijs 

 length ; but the length of the dyke is the same as the length of the 

 fluid which it supports ; consequently, the weight is very properly 

 represented by the area of the section and the specific gravity of the 

 material ; thus we have 



w as'. (147). 



But according to the writers on mensuration, the area of the trape- 

 zoid A BCD, is equal to the sum of the parallel sides AB and DC, drawn 

 into half the perpendicular distance AK or BL ; hence we have 



a=(AB -f- DC)XjAK, 



but by the foregoing notation, it is 



A B b (c -{- e) ; 

 consequently, by addition, we have 



A B -f DC = 26 (c + e) ; 

 therefore, the area of the section is 



a = ID (26 c e); 



let this value of a be substituted instead of it in the equation marked 

 (147), and it becomes 



w =. |DS' (26 c e) ; 



but the weight of the dyke is equivalent to the force whose momentum, 

 combined with that of the vertical pressure of the fluid, counterpoises 

 the momentum of the horizontal pressure, which force we have repre- 

 sented by F ; hence we have 



F IDS' (26 c e\ 

 and the momentum of this force, is 



IF M zz JD Is' (26 c e), 



where / denotes the lever whose length is equal to the distance 

 between the fulcrum, or centre of motion at c, and the vertical line 

 passing through the centre of gravity of the section A BCD ; conse- 

 quently, in the case of an equilibrium, we have 



mi^.m' -j- M, 

 and this, by restoring the analytical values, becomes 



Lsd* = s3d(tib 3) -{- ID/*' (26 c e). (148). 



209. This is the general equation which includes all the cases of 

 rectilinear sloping embankments, but it has not yet obtained its 

 ultimate form ; for the value of I has still to be expressed in terms 

 of the sectional dimensions, and in order to this, a separate investi- 

 gation becomes necessary. 



o2 



