110 OF FLUID PRESSURE UPON THE INTERIOR OF CONICAL VESSELS. 



draw B t and D t, which will be intersected by the straight lines, c m 

 and \n iu the points r and s; then are r and s respectively the 

 centres of gravity of the triangles ABC and ADC. 



Join the points r and s, by the straight line rs, intersecting mn in 

 the point G ; then it is obvious, that the common centre of gravity of 

 the triangles ABC and ADC, (which coincides with that of the trapezoid 

 ABCD), must occur in the line rs, which joins their respective centres. 



Now, because the trapezoid A B c D is symmetrically situated with 

 respect to the axis mn, it follows, that its centre of gravity must occur 

 in that line ; but we have shown above, that it also occurs in the line 

 rs, it consequently must be situated in the point G, where these lines 

 intersect one another ; hence, the centre of gravity of the surface of 

 the conic frustum occurs at the point G, and m G is its perpendicular 

 depth below AB the upper surface of the fluid. 



Put A =. the area of the lower base or bottom of the vessel, whose 



diameter is DC, 

 P the pressure perpendicular to its surface, or the weight of a 



quantity of fluid equal to the inscribed cylinder, 

 Dzzmw, the axis of the frustum, or the perpendicular depth of 



the centre of gravity of the bottom, 



a zz: the area of the curve surface of the vessel or conic frustum, 

 p zz: the pressure perpendicular to the curve surface, 

 d zz: m&, the perpendicular depth of its centre of gravity, 

 3 zz: DC, the diameter of the lower base or bottom of the vessel, 

 ft zz: AB, the diameter of the top or upper base, 

 w the weight, and s the specific gravity of the contained fluid. 



Then, by the principles of mensuration, the area of the lower base 

 of the conic frustum, or the bottom of the vessel on which the fluid 



presses, is 



Azzr.78542 2 , 



and consequently, the pressure upon it, is 



Pzz:.7854j 2 Ds. (82). 



This equation, having 5 2 instead of /3 2 , is the same as that which we 

 obtained for the pressure on the bottom in the preceding case, when 

 the greater base of the vessel was downwards ; it therefore follows, 

 since our notation is adapted to the same parts of the vessel, that not- 

 withstanding the inversion, the pressure on the curved surface of the 

 conic frustum, will still be expressed as in the equation marked (73) ; 

 consequently, we have 



P :p: :.7854$ 2 ns:: 1.5708 (/3-f 2)ds VD' 



