A Treatise on the Theory of Screws

223] FREEDOM OF THE FOURTH ORDER. 233 momental ellipsoid; every rigid body which fulfils nine conditions will belong to this family. If an impulsive wrench applied to a member of this family cause it to twist about a screw 6, then the same impulsive wrench applied to any other member of the same family will cause it likewise to twist about 0. If we added the further condition that the masses of all the members of the family were equal, then it would be found that the twist velocity, and the kinetic energy acquired in consequence of a given impulse, would be the same to whatever member of the family the impulse were applied (§§ 90, 91). 223. Quadratic n-systems. We have always understood by a screw system of the nth order or briefly an '/i-systcm, the collection of screws whose co-ordinates satisfy a certain system of 6— n linear homogeneous equations. We have now to introduce the conception of a screw system of the nth order and second degree or briefly a quadratic n-system (n<Q). By this expression we are to understand a collection of screws such that their co-ordinates satisfy 6 — n homogeneous equations; of these equations 5 — n, that is to say, all but one are linear; the remaining equation involves the co-ordinates in the second degree. Let 0j, ...,06 be the co-ordinates of a screw belonging to a quadratic n-system. We may suppose without any loss of generality that the 5 — n linear equations have been transformed into 0n+2 = O; 0n+8 = O; ... 00 = O. The remaining equation of the second degree is accordingly obtained by equating to zero a homogeneous quadratic function of 0i ■ • • 0n+i • We express this equation which characterizes the quadratic ft-system as Uo = O. All the screws whose co-ordinates satisfy the 5—n linear equations must themselves form a screw system of the 6 — (5 — n) = (n + l)th system. This screw system may be regarded as an enclosing system from which the screws are to be selected which further satisfy the equation of the second degree Ue = 0. The enclosing system comprises the screws which can be formed by giving all possible values to the co-ordinates 0lt ...,0n+1. Of course there may be as many different screw systems of the nth order and second degree comprised within the same enclosing system as there can be different quadratic forms obtained by annexing coefficients to the several squares and products of n + 1 co-ordinates. If n = 5, the enclosing system would consist of every screw in space.