302 Hydrostatics. 



ing from the pressure of the fluid passes ; the second to find 

 what becomes of the horizontal forces. 



(1.) It will be seen that the vertical effort must pass through 

 the centre of gravity of the portion of fluid displaced. For, if 

 we imagine this portion decomposed into an infinite number of 

 vertical filaments, the vertical effort which the fluid exerts upon 

 each filament, is expressed by the weight of a portion of fluid 



272. equal to this filament ; consequently, to find the distance of the 

 the resultant from any vertical plane, it is necessary to multiply 

 the mass of each filament, considered as of the same nature with 

 this fluid, by its distance from this plane, and to divide by the 

 sum of the filaments. But this is precisely the course to be 



76. pursued, in order to find the distance of the centre of gravity of 

 the portion of fluid displaced ; therefore the vertical effort of a 

 fluid upon a body immersed in it, passes always through the 

 centre of gravity of the portion of fluid displaced, which may be 

 called the centre of buoyancy. 



421. (2.) We proceed now to consider the horizontal forces 

 above referred to. Representing always the solid stratum by 

 figure 203, if through the sides 6, 6 c, &c., of the inferior sec- 

 tion, we suppose vertical planes to pass terminating in the supe- 

 rior section ; these planes will form the contour of a prism whose 

 altitude is that of the stratum ; and each face of this prism will 



417. express by the extent of its surface the value of the horizontal 

 force perpendicular to it. But, since all these faces are of the 

 same altitude, their surfaces will be as their bases a 6, b c, &c., 



e^>m. consequently the horizontal forces are to each other as the sides 

 a 6, b c, &;c. Moreover, at whatever point of these faces they 

 are applied, as these faces are of an altitude infinitely small, the 

 horizontal forces may be considered as applied each in the hor- 

 izontal plane a b c d ef, perpendicularly to the middle of the side 

 which s,erves as a base to the corresponding face of the prism in 

 question. I say to the middle, since it will readily be seen that 

 the resultant of the pressures exerted upon the surface of any one 

 of the trapezoids which form the surface of the stratum, must 

 pass through some point of the line joining the middle points of 

 the two parallel sides, and that, accordingly, the horizontal force 

 obtained by decomposing this resultant, must meet the line join- 



