A Treatise on the Theory of Screws

242 THE THEORY OF SCREWS. [227, belonging to A, then the body will commence to twist about the screw 6, of which t] is the polar with respect to the quadratic n-system composed of the imaginary screws about which the body would twist with zero kinetic energy. If a rigid body which has freedom of the nth order be displaced from a position of stable equilibrium under the action of a system of forces by a twist of given amplitude about a screw 3, of which the co-ordinates referred to the n principal screws of the potential are 31>... 3n, then the potential energy of the new position may, as we have seen (§ 103) be expressed by ^3^ + ... +Vni3ni. If this expression be equated to zero, it denotes a quadratic «-system, which is of course imaginary. We may term it the potential quadratic n-system. The potential quadratic «-system possesses a physical importance in every respect analogous to that of the kinetic quadratic n-system: by reference to (§ 102) the following theorem can be deduced. If a rigid body be displaced from a position of stable equilibrium by a twist about a screw 6, then a wrench acts upon the body in its new position on a screw which is the polar of 3 with respect to the potential quadratic n-system. The constructions by which the harmonic screws were determined in the case of the second and the third orders have no analogies in the fourth order. We shall, therefore, here state a general algebraical method by which they can be determined. Let U = 0 be the kinetic quadratic n-system, and V = 0 the potential quadratic n-system, then it follows from a well-known algebraical theorem that one set of screws of reference can in general be found which will reduce both U and V to the sum of n squares. These screws of reference are the harmonic screws. We may here also make the remark, that any quadratic n-system can generally be transformed in one way to the sum of n square terms with co-reciprocal screws of reference; for if Ue and pe be transformed so that each consists of the sum of n square terms, then the form for the expression of p0 (§ 38) shows that the screws are co-reciprocal. 228. On the degrees of certain surfaces. We have already had occasion (§ 210) to demonstrate that the general condition that two screws shall intersect involves the co-ordinates of each