320 BOOK OF MATHEMATICAL PROBLEMS. 



axis is parallel to one of the principal axes at the centre of gravity; 

 prove that the fixed point must lie on a certain hyperbola, one 

 asymptote of which is the given principal axis. 



1527. A free rigid body is at a certain moment in a state of 

 rotation about an axis through its centre of gravity, when another 

 point of the body suddenly becomes fixed; determine the new 

 instantaneous axis, and prove that there are three directions of 

 the former instantaneous axis for which the new axis will be in 

 the same direction, and that these three directions are along con- 

 jugate diameters of the central ellipsoid at the centre of gravity. 



1528. A rigid body is in motion under the action of no 

 forces, its centre of gravity being at rest, and the instantaneous 

 axis describes a plane in the body ; prove that if a point in that 

 diameter of the central ellipsoid which is conjugate to this plane 

 be suddenly fixed, the new instantaneous axis will be parallel to 

 the former. 



1529. Two equal uniform rods AB, BC, freely jointed at B 

 and in one straight line, are moving uniformly in a direction at 

 right angles to their length on a smooth horizontal table, when A is" 

 suddenly fixed; prove that the initial angular velocities of the 

 rods are in the ratio 3 : 1, that the least subsequent angle 

 between them is 



and that when next in a straight line their angular velocities are 

 as 1 : 9. 



1530. Three equal uniform rods AB, BC^CD, freely jointed 

 at B and C, are lying in one straight line on a smooth table, when 

 a blow is applied at the middle point at right angles to the rods; 

 prove that 



d& _ o) 



eft ~,y(l+sin 8 0)' 



, 



