300 Hydrostatics. 



the trapezoid ABCD infinitely small compared with the sides 

 AB and CD, EF which is equal to CD, may be taken instead 



or) I r<r\ 



of AB and also instead of CD, so that HI X - - reduces 



itself to HI X ^j? = HI X EF, which is the surface of the 



rectangle ECDF-, we have, therefore, 



p : q : r : : ABCD X GG' : ECDF X GG' : AFEB X GG'. 



But we have supposed the force p expressed by ABCD X GG', 

 hence the force q will be expressed by ECDF X GG', and the 

 force r by AFEB X GG'. 



Since a triangle is simply a trapezoid, one of whose parallel 

 sides is zero, the same results are applicable to a triangle. 



Suppose now perpendiculars let fall from the angles A, D, C, 

 B, upon the plane XZ. These perpendiculars may be consider- 

 ed as the edges of a truncated prism, the horizontal base being 

 AFEB, and the inclined base ABCD. Now as AB, CD, are 

 supposed to be infinitely near to each other, the bulk of this 

 prism may be regarded as not differing from that of a prism of 

 the same base, and whose altitude is GG 7 ; but this last has for 

 its expression AFEB X GG', which is precisely that just found 

 for the vertical force r ; therefore this force has also for its ex- 

 pression, the bulk of the truncated prism, whose inclined base is 

 ABCD and whose horizontal base is the projection of ABCD 

 upon the horizontal plane XZ. 



418. Let any solid be divided into an infinite number of hor- 

 Fig.203. i zonta l strata ABDE abde, and suppose that at the centre of 

 gravity of the surface of each trapezoid of which the surface of 

 the perimeter of this stratum is composed, forces are applied, 

 represented each by the surface of the corresponding trapezoid 

 multiplied by the distance of the centre of gravity from a hori- 

 zontal plane XZ, These forces will represent the pressure ex- 

 416 erted by a heavy fluid upon the interior surface of the stratum 

 ABDE a b d e of a vessel in which this fluid is contained ; they 

 will also represent the pressure exerted by a similar fluid upon 

 the exterior surface of a solid immersed in this fluid. Now we 

 have seen that these forces being decomposed each into two 



