in 



42. A solid sector of a sphere hangs from a point in its 

 ;!.ir rim with its axis horizontal, find its vertical an 



Result. The cosine of the semi-vertical anirlc is f. 



43. Find the centre of gravity of the solid 1 l.y 

 the revolution of a semicircle about a straight line perp. -n- 

 clicular to the diameter, and which does not meet the semi- 

 circle. 



Result. Distance from the plane generated by the diameter 



\r 



44. A is a point in the generating line of a right cylinder 

 on a circular base, and B, V are two others in the generating 

 line diametrically opposite. The cylinder is bisected by a 

 plane ABC, and one of the semicylinders is cut by two planes 

 at right angles to ABC, passing through AB and A C. ^ 

 that if the solid ABC be placed with its convex side on a 

 horizontal plane, the plane ABC will be inclined to the hori- 

 zon at an angle tan" 1 (^TT), when there is equilibrium. 



45. A solid cone is cut by two planes perpendicular to 

 the same principal section, one through its axis, and the 

 other parallel to a slant side; find the limiting value of the 

 vertical angle of the cone, that the piece cut out may rt 

 its curved surface on a horizontal plane. 



Result. The cosine of the vertical angle must not be 

 greater than . 



40. A ouadrant of a circle revolves round one of its 

 extreme ractii through an angle of 30; find the centre of 

 gravity of the solid traced out, the density being supposed 

 to vary as the distance from the centre. 



Results. x = r ; y = - - (2 - V3) ; z = r . The axis of x 

 > o o 



is supposed to coincide with the initial position of the revolv- 

 ing radius. 



