DYNAMICS OF A RIGID BODY. 1 13 



formed about a screw 0, which can very simply be con- 

 structed by the ellipse of the potential. Determine the 

 screw ^ (on the cylindroid of freedom) which is recipro- 

 cal to 7j ( 28), then 0, and the required screw 0, are 

 parallel to a pair of conjugate diameters of the ellipse 

 of the potential. 



The common conjugate diameters of the pitch conic, 

 and the ellipse of the potential, are parallel to the two 

 screws on the cylindroid, which we have designated the 

 principal screws of the potential ( 71). 



When a body is displaced from its position of equili- 

 brium by a small wrench upon a principal screw of the 

 potential, then the body moves to the new position 

 which is required in its altered circumstances by a small 

 twist about the same screw. 



104. Harmonic Screws. The common conjugate 

 diameters of the ellipse of inertia, and the ellipse of 

 the potential, are parallel to the two harmonic screws 

 on the cylindroid ( 74). This is evident, because the 

 pair of screws thus determined are conjugate screws 

 both of inertia and of the potential. 



If the body be displaced by a twist about one of the 

 harmonic screws, and be then abandoned to the influ- 

 ence of the forces, the body will continue for ever to 

 perform twist oscillations about that screw. 



If the ellipse of inertia, and the ellipse of the poten- 

 tial, be similar, and similarly situated, then every screw on 

 the cylindroid of freedom will be an harmonic screw. 



105. Exceptional Case. We have now to consider 

 the modifications which the results we have arrived at 

 undergo when the cylindroid becomes illusory in the 

 case considered ( 94). 



Suppose that 4 and were a pair of conjugate screws 

 of inertia on the straight line about which the body was 

 free to rotate and slide independently. Then taking 



I 



