336 BOOK OF MATHEMATICAL PROBLEMS. 



If the cone be allowed to sink freely into the fluid, its vertex 

 being initially at the surface, and sinks till just immersed, the 

 density of the cone is to the density of the fluid at the vertex of 

 the cone in its lowest position as 1 : 30. 



1590. A semicircular tube of fine bore whose plane is 

 vertical contains a quantity of fluid which subtends a given angle 

 at the centre ; a given heavy particle just fitting the tube is let 

 fall from the extremity of a horizontal radius : find the impulsive 

 pressure at any point of the fluid. 



1591. A flexible inextensible envelope when filled with fluid 

 has the form of a paraboloid, whose axis is vertical and vertex 

 downwards and whose height is equal to five-eighths of the latus 

 rectum ; determine where the tension of the envelope along the 

 meridian is greatest. 



1592. Fluid without weight is contained in a thin flexible 

 envelope in the form of a surface of revolution, and the tensions 

 of the envelope at any point along and perpendicular to the 

 meridian are equal : prove that the surfoce is a sphere. 



1593. A quantity of homogeneous fluid is contained between 

 two parallel planes and is in equilibrium in the form of a cylinder 

 of radius b under a pressure -or ; that portion of the fluid which 

 lies within a distance a of the axis being suddenly annihilated, 

 prove that the initial pressure at any point at a distance r from 

 the axis is 



log r log a 

 log b log a * 



1594. A thin hollow cylinder of length h closed at one end, 

 and fitted with an air-tight piston is placed mouth downwards in 

 fluid. The weight of the piston is equal to that of the cylinder, 

 the height of a cylinder of equal weight and radius formed of the 

 fluid is a, the height of fluid which measures the atmospheric 

 pressure is c, and the air enclosed in the cylinder would just fill it 



