197] FREEDOM OF THE THIRD ORDER. 195 



diameters common to the two ellipses in which the plane conjugate to the 

 vertical axis in the momental ellipsoid cuts the momental ellipsoid and the 

 cylinder. These three lines are the three harmonic axes. 



As to that vertical axis which appears to be one of the harmonic 

 axes, the time of vibration about it would be infinite. The three har 

 monic screws which are usually found in the small oscillations of a body 

 with freedom of the third order are therefore reduced in the present case 

 to two, and we have the following theorem : 



A rigid body which is free to rotate about a fixed point is at rest under 

 the action of gravity. If a plane S be drawn through the point of suspension 

 0, conjugate to the vertical diameter 01 of the momental ellipsoid, then the 

 common conjugate diameters of the two ellipses in which 8 cuts the momental 

 ellipsoid, and a circular cylinder whose axis is 01, are the two harmonic axes. 

 If the body be displaced by a small rotation about one of these axes, the 

 body will continue for ever to oscillate to and fro upon this axis, just as if the 

 body had been actually constrained to move about this axis. 



To complete the solution for any initial circumstances of the rigid body, 

 a few additional remarks are necessary. 



Assuming the body in any given position of equilibrium, it is first to be 

 displaced by a small rotation about an axis OX. Draw the plane containing 

 01 and OX, and let it cut the plane S in the line OF. The small rotation 

 around OX may be produced by a small rotation about 01, followed by a 

 small rotation about OF. The effect of the small rotation about 01 is 

 merely to alter the azimuth of the position, but not to disturb the equi 

 librium. Had we chosen this altered position as that position of equilibrium 

 from which we started, the initial displacement would be communicated by a 

 rotation around F. We may, therefore, without any sacrifice of generality, 

 assume that the axis about which the initial displacement is imparted lies 

 in the plane S. We shall now suppose the body to receive a small angular 

 velocity about any other axis. This axis must be in the plane S, if small 

 oscillations are to exist at all, for the initial angular velocity, if not capable 

 of being resolved into components about the two harmonic axes, will have a 

 component around the vertical axis 01. An initial rotation about 01 would 

 give the body a continuous rotation around the vertical axis, which is not 

 admissible when small oscillations only are considered. 



If, therefore, the body performs small oscillations only, we may regard 

 the initial axis of displacement as lying in the plane S, while we must have 

 the initial instantaneous axis in that plane. The initial displacement may 

 be resolved into two displacements, one on each of the harmonic axes, and 



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