168] THE GEOMETRY OF THE CYLINDROID. 169 



to consider two conies connected with the cubic, viz. the reciprocal conic, 

 which is the envelope of reciprocal chords, and the inertia conic, which is 

 the envelope of chords of conjugate screws of inertia. We must provide 

 a means of discriminating the two tangents from a point P on the cubic 

 to either conic ; any ray, of course, cuts the cubic in three points, of which 

 two possess the characteristic relation. If P be one of these two, we may 

 call this tangent the odd tangent. The other tangent will have, as its 

 significant points, the two remaining intersections ; leaving out P, we can 

 then proceed, as follows, to determine the impulsive screw corresponding to 

 P as the instantaneous screw : 



Draw the odd tangent from P to the inertia conic, and from the con 

 jugate point thus found draw the odd tangent to the reciprocal conic. The 

 reciprocal point Q thus found is the impulsive screw corresponding to P as 

 the instantaneous screw. 



In general there are four common tangents to the two conies. Of these 

 tangents there is only one possessing the property, that the same two of its 

 three intersections with the cubic are the correlative points with respect 

 to each of the conies. These two intersections are the principal screws of 

 inertia. 



To determine the small oscillations we find the potential conic, the 

 tangents to which are chords joining two conjugate screws of the potential 

 ( 100). The two harmonic screws are then to be found on one of the two 

 common tangents to the two conies. It can be shown that both the inertia 

 conic and the potential conic will, like the reciprocal conic, have triple 

 contact with the cubic. 



