328 BOOK OF MATHEMATICAL PROBLEMS. 



1548. A circular tube of uniform bore whose plane is vertical 

 i> half filled with equal volumes of four fluids which do not mix 

 and whose densities are as 1 : 4 : 8 : 7; prove that the diameter 

 joining the free surfaces will make an angle tan" 1 2 with the 

 vertical. 



1549. A triangular lamina ABC right-angled at C is attached 

 to a string at A, and rests with the side AC vertical and half its 

 length immersed in fluid : prove that the density of the fluid is to 

 that of the lamina :: 8 : 7. 



1550. A lamina in the form of an equilateral triangle sus- 

 pended freely from an angular point rests with one side vertical 

 and another side bisected by the surface of a heavy uniform fluid : 

 prove that the density of the lamina is to that of the fluid as 

 15 : 16. 



1551. A hollow cone filled with fluid is suspended freely 

 from a point in the rim of the base : prove that the total pres- 

 sures on the curve surface and on the base in the position of rest 

 are in the ratio 



1 + 11 sin 2 a : 12 sin 3 a, 



2a being the angle of the cone. 



1552. A tube of small bore in the form of an ellipse is half 

 filled with equal volumes of two given fluids which do not mix ; 

 find the inclination of its axes to the vertical in order that the 

 free surfaces of the fluids may be at the ends of the minor axis. 



1553. A hemisphere is filled with fluid and the surface is 

 divided by horizontal planes into n portions on each of which the 

 whole pressure is the same ; prove that the depth of the r th of 

 these planes is to the radius of the hemisphere as Jr : Jn. 



1554. A hemisphere is just filled with a uniform heavy fluid 

 and the surface is divided by horizontal planes into n portions the 

 whole pressures on which are in a geometrical progression of ratio 

 k : prove that the depth of the r th plane is to the radius as 



