112 OF FLUID PRESSURE UPON THE INTERIOR OF CONICAL VESSELS. 



117. In order to compare the pressure on the bottom of the vessel, 

 with the weight of the fluid which it contains, we must again have 

 recourse to the principles of solid geometry ; from which we learn, 

 that the solid content of a conic frustum, whose diameters are denoted 

 by (3 and 5, and its perpendicular altitude by D, is 



and consequently, its weight becomes 



to = .2618 DS (0* + ft 3 + S 2 ) ; (85). 



therefore, by comparing this equation with that marked (82), we get 



. P : w : : 33* : (/3 2 + /3 3 + a 2 ). (86). 



Again, if we compare the equations marked (73 and (85) with one 

 another, we shall have 



p : w :: 1.5708 (0 + 3) ds v 'D* -*%(& 3) : .2618 Ds(/3 2 + /33 + J') ; 

 if therefore, we expunge the common quantities, from the third and 

 fourth terms of the above analogy, and in the third term, substitute the 

 value of d as it is expressed in the equation (83), then we shall obtain 



p : w : : 2(0 + 2 3) <J D* -f i (ft *> : (/3 2 + ft 3 + *). 

 When 5 vanishes, or when the vessel becomes a complete cone with 

 its vertex downwards, the preceding analogy gives 



p : w : : 2 ^ D* + |/3 2 : /3. 



In complying with the conditions of the 20th problem, the fore- 

 going investigation has been conducted on the supposition, that the 

 vessel in question is in the form of the frustum of a cone ; but the 

 attentive reader will readily perceive, that the same mode of procedure 

 will apply to the frustum of any other regular pyramid, and the result- 

 ing formulae will partake of similar forms and combinations, differing 

 only in so far as depends upon the constant numbers which express 

 their respective areas and solidities ; it is therefore unnecessary to 

 pursue the inquiry further, taking it for granted, that by a careful 

 perusal of what has been done above, no difficulty will be met with 

 in applying the same principles to any other case of form or condition 

 that is likely to occur. 



COUOL. 1. By the preceding investigation, then, and the formulae 



arising from it, we learn, that by causing the sides of a vessel, which 



is filled with an incompressible and non-elastic fluid, to converge or 



diverge from the extremities of the base, supposed to be horizontal : 



The pressure on the base, may be greater or less than the 



weight of the fluid which the vessel contains, in any propor- 



tion whatever. 



