280 On the Inertia of a Free or Constrained Rigid Body. 



We shall illustrate the existence of these principal screws 

 of inertia by pointing them out in the particular cases of 

 freedom of the second and third orders. When the body has 

 freedom of the second order, the screw-system is a cylindroid. 

 All the generators of a cylindroid are parallel to a plane ; and 

 by the anharmonic ratio of four generators is to be understood 

 the anharmonic ratio of four parallel rays drawn through any 

 point. Now it can be shown that the anharmonic ratio of four 

 instantaneous screws on the cylindroid is equal to the anhar- 

 monic ratio of the four corresponding impulsive screws. When, 

 therefore, three impulsive screws and the three corresponding 

 instantaneous screws are known, the instantaneous screw 

 corresponding to any impulsive screw is at once determined 

 by geometry. The double rays of the two equianharmonic 

 systems must of course be parallel to the two principal screws 

 of inertia on the cylindroid ; and thus the problem of finding 

 the principal screws of inertia for freedom of the second order 

 is completely solved. 



In the case of freedom of the third order, the three principal 

 screws of inertia can also be completely determined by geo- 

 metry. For this purpose it is necessary to construct a certain 

 ellipsoid, which is defined by the following theorem. 



The kinetic energy of a rigid body when twisting with a 

 given twist velocity about any screw of a screw-system of the 

 third order is proportional to the inverse square of the paral- 

 lel diameter of a certain ellipsoid. 



If this ellipsoid be made concentric with the pitch-quadric, 

 it will be possible to draw a triad of common conjugate dia- 

 meters to the two surfaces ; and the required principal screws of 

 inertia are the three screws of the complex which are parallel 

 to these conjugate diameters. 



In that special case of freedom of the third order in which 

 a body is rotating about a fixed point, then the general pro- 

 perty of the three principal screws of inertia degrades to the 

 well-known property of the principal axes. It will be ob- 

 served that the theory here propounded may be considered to 

 generalize the property of the principal axes into a general 

 property for freedom of the third order, and then further into 

 a property for freedom of any order. 



We conclude by pointing out the six principal screws of 

 inertia of a perfectly free rigid body. They are found as 

 follows : — Draw the three principal axes, A, B, C, through the 

 centre of gravity, and let a, b, c be the radii of gyration; then 

 two screws on A with the pitches +a, —a, and two similar 

 screws on B and on C, constitute the six principal screws of 

 inertia. 

 Dundnk, Co, Dublin, August 1878. 



