334 BOOK OF MATHEMATICAL PROBLEMS. 



point by a force varying as the distance; find the pressure at any 

 point. 



1581. Water is contained in a vessel having a horizontal 

 base, and a right cone is supported, partly by the water and partly 

 by the base on which the vertex rests : prove that, for stable equi- 

 librium, the depth of the fluid must be greater than h t](m cos 2 a), 

 m being the number measuring the specific gravity of the cone, 

 h the length of the axis, and 2a the vertical angle. 



1582. A solid paraboloid is divided into two parts by a plane 

 through the axis, and the parts are united by a hinge at the 

 vertex; the system is placed in a heavy uniform fluid with its axis 

 vertical and vertex downwards, and floats without separation of 

 the parts : prove that the ratio of the density of the solid to that 

 of the fluid must be greater than # 4 , x being given by the equation 



3hx*=7a(l-x) t 



where h, 4a are the lengths of the axis and the latus rectum re- 

 spectively. 



1583. A right cone is floating with its axis vertical and 

 vertex downwards in a fluid whose density varies as the depth : 

 prove that, for stable equilibrium, 



^5 



2a being the vertical angle, p the density of the cone, and o- the 

 density of the fluid at a depth equal to the height of the cone. 



1584. A uniform rod rests in a position inclined to the ver- 

 tical with half its length immersed in a fluid, and can turn freely 

 about a point in it at a distance equal to one-sixth of its length 

 from the lower end : compare the densities of the rod and fluid, 

 and prove that the equilibrium is stable. 



1585. A uniform rod is moveable about one extremity which 



