211] PLANE REPRESENTATION OK THE THIRD ORDER. 217
point of contact, it follows that every screw on the cylindroid must be at
right angles to the direction corresponding to the point of contact. The
co-ordinates of the point of contact must therefore be proportional to the
direction cosines of the nodal line of the cylindroid.
If the body be in equilibrium under the action of a conservative system
of forces, then there is a conic (analogous to the conic of inertia) which
denotes the locus of screws about which the body can be displaced to a
neighbouring position, so that even as far as the second order of smal]
quantities no energy is consumed. The vertices of the triangle self-conjugate
both to this conic and the conic of inertia correspond to the harmonic
screws about which, if the body be once displaced, it will continue to
oscillate.