106 



4. WHEN THE VESSEL ASSUMES THE FORM OF A TRUNCATED CONE, 

 THE BASE OF WHICH IS ALSO THE BOTTOM OF THE VESSEL, AND 

 ITS AXIS PERPENDICULAR TO THE HORIZON. 



PROBLEM XX. 



115. If a vessel in the form of the frustum of a cone, be 

 filled with an incompressible and non-elastic fluid, and have its 

 axis perpendicular to the horizon : 



It is required to compare the pressure on the bottom of the 

 vessel with that upon its curved surface, and also with the 

 weight of the fluid which it contains, both when the sides of 

 the vessel converge, and when they diverge from the extremities 

 of the bottom. 



Let ABCD represent a vertical section of a vessel in the form of the 

 frustum of a cone, and filled with an incom- 

 pressible and non-elastic fluid whose horizontal 

 surface is AB ; produce AB both ways, to any 

 convenient distance, and through D and c the 

 extremities of the bottom diameter, draw Da 

 and c b respectively perpendicular to D c, and 

 meeting AB produced in the points a and b; 

 then is abcv the vertical section, passing 

 along the axis of the cylinder which circumscribes the conic frustum. 



Bisect AB and DC respectively in the points m and n, and draw 

 the straight line mn\ then, because the figure ABCD is symmetrical 

 with respect to the axis mn, it follows, that mn bisects the figure 

 or trapezoid ABCD, and consequently passes through its centre of 

 gravity. 



Draw the diagonal AC, dividing the figure ABCD into the two 

 triangles ABC and ADC ; then it is manifest, that the common centre 

 of gravity of the two triangles, and that of the trapezoid constituted 

 by their sum, must occur in one and the same point ; therefore, bisect 

 the diagonal A c in the point t, and draw A n and D t intersecting each 

 other in the point r, and c m, B t intersecting in s ; then are r and s 

 the centres of gravity of the triangles ADC and ABC; draw rs inter- 

 secting mn in G, and G will be the centre of gravity of the trapezoid 

 ABCD. 



Now, it is demonstrated by the writers on mechanics, that the 

 centre of gravity of the surface of a conic frustum : 



