211] PLANE REPRESENTATION OF THE THIRD ORDER. 215 



We have first to draw the conic of which the equation is 



This conic is of course imaginary, being in fact the locus of screws about 

 which, if the body were twisting with the unit of twist velocity, the kinetic 

 energy would nevertheless be zero. If two points 0, (ft are conjugate with 

 respect to this conic, then 



The screws corresponding to 6 and $ are then what we have called conjugate, 

 screws of inertia. 



This conic is referred to a self-conjugate triangle, the vertices of 

 which are three conjugate screws of inertia. There is one triangle self- 

 conjugate both to the conic of zero pitch, and to the conic of inertia just 

 considered. The vertices of this triangle are of especial interest. Each pair 

 of them correspond to a pair of screws which are reciprocal, as well as being 

 conjugate screws of inertia. They are therefore what we have designated 

 as the principal screws of inertia ( 87). They degenerate into the principal 

 axes of the body when the freedom degenerates into the special case of 

 rotation around a fixed point. 



When referred to this self-conjugate triangle, the relation between the 

 impulsive point and the corresponding instantaneous point can be expressed 

 with great simplicity. Thus the impulsive point &amp;lt;, whose co-ordinates are 



#iWi 2 -5- Pi ; 0*uf -j- p 2 ; 3 3 a -r p 3 , 



corresponds to the instantaneous point whose co-ordinates are 6 lt 2 , 3 . The 

 geometrical construction is sufficiently obvious when derived from the 

 theorem thus stated. 



If &amp;lt;f&amp;gt; denote an impulsive screw, and 6 the corresponding instantaneous screw, 

 then the polar of &amp;lt;f&amp;gt; with regard to the conic of zero pitch is the same straight line 

 as the polar of 6 with regard to the conic of inertia. 



If H be the virtual coefficient of two screws 6 and 77, then 



It follows that the locus of the points which have a given virtual coefficient 

 with a given point is a conic touching the conic of infinite pitch at two 

 points. If -v/r be the screw whose polar with regard to the conic of infinite 

 pitch is identical with the polar of 77 with regard to the conic of zero pitch, 

 then all the screws 6 which have a given virtual coefficient with 77 arc 

 equally inclined to ^. It hence follows that all the screws of a three- 

 system which have a given virtual coefficient with a given screw are parallel 



