OF FLUID PRESSURE UPON THE INTERIOR OF CONICAL VESSELS. Ill 



or by expunging the common quantities, we get 



P :p : : 3*D : 2(/3 + 3) dji>* + J(/3 3)'. 



But the writers on mechanics have demonstrated, that the depth of 

 the centre of gravity of the trapezoid A BCD, below the horizontal line 

 AB, is expressed as follows, viz. 



' (83). 



let therefore, this value of d be substituted instead of it in the above 

 analogy, and we shall obtain 



P : p : : 33* : 2 (ft + 23) V * + I (ft V- 



If , the diameter of the bottom or lower base should vanish, the 

 vessel becomes a complete cone with its vertex downwards, in which 

 case, the value of d as expressed in the equation marked (83), is 



Let this value of d be substituted instead of it, in the equation 

 marked (73), and suppose 3 to vanish ; then, the pressure on the con- 

 cave surface of a conical vessel with its vertex downwards, becomes 



p = .5236 ft D s V D 9 + J r /P. (84). 



The solid content of the inscribed cylinder, of which the vertical 

 section passing along the axis is a6co, becomes 



c'=r.78543 2 D, 



and as we have already stated, its weight is proportioned to its mag- 

 nitude drawn into the specific gravity ; hence we have 



/ = . 7854 3*Ds; 



but this is the same expression which indicates the pressure on the 

 bottom, as exhibited in the equation marked (82) ; hence it follows, 

 that the pressure on the bottom or lower base of the conic frustum, 

 when the sides diverge from the extremities of its diameter, 



Is equal to the weight of a column of the fluid, of the same 

 magnitude as the cylinder inscribed in the conic frustum. 



But the solid content of the inscribed cylinder, and consequently 

 its weight, is manifestly less than the content of the vessel ; hence we 

 infer, that when the sides of the vessel diverge from the extremities of 

 the diameter of its bottom, the pressure on the bottom is less than 

 the weight of the fluid which it contains, the remaining weight being 

 supported by the resistance of the diverging sides. 



