144 DYNAMICS OF A RIGID BODY. 



be infinite, so we reject it. The three harmonic 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 O, con- 

 gate to the vertical diameter OI of the momental ellip- 

 soid, then the common conjugate diameters of the two 

 ellipses in which S cuts the momental ellipsoid, and 

 a circular cylinder whose axis is OI, are the two har- 

 monic axes. If the body be displaced by a small rota- 

 tion 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 circum- 

 stances of the rigid body, a few additional remarks are 

 necessary. 



Assuming the body in any given position of equili- 

 brium, it is first to be displaced by a small rotation about 

 an axis OX. Draw the plane containing OI and OX, 

 and let it cut the plane S in the line OY. The small 

 rotation around OX may be produced by a small rota- 

 tion about OI, followed by a small rotation about OY. 

 The effect of the small rotation about OI is merely to alter 

 the azimuth of the position, but not to disturb the equili- 

 brium. Had we chosen this altered position as that 

 position of equilibrium from which we started, the ini- 

 tial displacement will be communicated by a rotation 

 around OY. We may, therefore, without any sacrifice 

 of generality, assume that the axis about which the 

 initial displacement is imparted lies in the plane *$*. We 

 shall now suppose the body to receive a small angular 



