A Treatise on the Theory of Screws

211] PLANE REPRESENTATION OF THE THIRD ORDER. 215 We have first to draw the conic of which the equation is + uifl + u?0,? = 0. 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, </> are conjugate with respect to this conic, then u^Øfg + u202<f>2 + ws"03^>3 = 0. The screws corresponding to 0 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 whose co-ordinates are Ogi^-pi, ØM/^pi, Ø3us^p3, corresponds to the instantaneous point whose co-ordinates are 0lt 02 , 03. Ihe geometrical construction is sufficiently obvious when derived from the theorem thus stated. If denote an impulsive screw, and 0 the corresponding instantaneous screw, then the polar of </> with regard to the conic of zero pitch is the same straight line as the polar of 0 with regard to the conic of inertia. If II be the virtual coefficient of two screws 0 and g, then H2 (0? + 01 + 0f) = (pfiVi + P^W* + pApjf 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 -»/r be the screw whose polar with regard to the conic of infinite pitch is identical with the polar of g with regard to the conic of zero pitch, then all the screws 0 which have a given virtual coefficient with g are equally inclined to f. It hence follows that all the screws of a three- system which have a given virtual coefficient with a given screw are parallel