330 BOOK OF MATHEMATICAL PROBLEMS. 



placed nnder a receiver from which the air is removed and the 

 barometer then stands at -36 of an inch ; find the space which 

 would be occupied by a given volume of the atmosphere if it were 

 deprived of its vapour without changing its pressure or tempe- 

 rature. 



1561. In Hawksbee's air-pump, when the w th stroke is half 

 completed the machine is kept at rest ; find the difference of the 

 tensions of the two piston rods. 







1562. In Smeaton's air-pump, find the position of the piston 



at that moment of the n ih stroke when the upper valve begins to 

 open. 



1563. If A, B be the volumes of the receiver and barrel of an 

 air-pump, p, <r the densities of atmospheric air and of the air in 

 the receiver respectively, and II the pressure of the atmosphere ; 

 the work done in slowly raising the piston through one stroke is 



gravity being neglected. 



1564. A portion of a right circular cone cut off by a plane 

 through the axis, and two planes perpendicular to the axis is 

 immersed in a uniform heavy fluid in such a manner that the 

 vertex of the cone is in the surface and the axis vertical : prove 

 that the resultant horizontal pressure on the curve surface passes 

 through the centre of gravity of the body immersed. 



1565. If it be assumed that the temperature of the atmo- 

 sphere in ascending from the Earth's surface decreases slowly by 

 an amount proportional to the height ascended, prove that the 

 equation connecting the pressure p and the density p at any height 

 will be of the form p = kp l+m , m being very small. 



1566. A right circular cylinder floats in a uniform heavy 



2 



fluid with its axis inclined at an angle tan" 1 = to the vertical, its 







