DYNAMICS OF A RIGID BODY. 1 1 1 



the cylindroid of freedom are parallel to a pair of conju- 

 gate diameters of the pitch conic. The construction is 

 therefore, as follows : Find the diameter A which is 

 conjugate, with respect to the ellipse of inertia to the 

 diameter parallel to the given screw 0. Next find the 

 diameter B which is conjugate to the diameter A with 

 respect to the pitch conic. The screw on the cylindroid 

 parallel to the line B thus determined is the required 

 screw rj. 



Two concentric ellipses have one pair of common 

 conjugate diameters. In fact, the four points of inter- 

 section form a parallelogram, to the sides of which 

 the pair of common conjugate diameters are parallel. 

 We can now interpret physically the common conjugate 

 diameters of the pitch conic, and the ellipse of inertia. 

 The two screws on the cylindroid parallel to these 

 diameters are conjugate screws of inertia, and they are 

 also reciprocal ; they are, therefore, the principal screws of 

 inertia, to which we have been already conducted ( 101). 



If the distribution of the material of the body bear 

 certain relations to the arrangement of the constraints, 

 we can easily conceive that the pitch conic and the 

 ellipse of inertia might be both similar and similarly 

 situated. Under these exceptional circumstances it 

 appears that every screw of the cylindroid would possess 

 the property of a principal screw of inertia. 



103. The Ellipse of the Potential. We are now to 

 consider another ellipse, which, though possessing many 

 useful mathematical analogies to the ellipse of inertia, is 

 yet widely different from a physical point of view. We 

 have introduced ( 72) the conception of the linear mag- 

 nitude # a , the square of which is proportional to the 

 work done in effecting a twist of given amplitude about 

 a screw a from a position of stable equilibrium under 

 the influence of a system of forces. We now propose 



