A Treatise on the Theory of Screws

104] HARMONIC SCREWS. 95 must be fulfilled; but this is precisely the number of arbitrary elements available in the selection of 3. We can thus conceive that for one or more particular screws 3, the two corresponding screws X and y are identical; and we shall now prove the following important theorem:— If a rigid body be displaced from a position of equilibrium by a twist about a screw 3, and if the evoked wrench tend to make the body commence to twist about the same screw 3, then if we call 3 an harmonic screw (§ 106), we assert that the number of harmonic screws is generally the same as the order of the screw system which defines the freedom of the rigid body. We shall adopt as the screws of reference the n principal screws of inertia. The impulsive screw, which corresponds to 3 as an instantaneous screw, will have for co-ordinates h^3A, ... hUlhn (§ 97), Pi Pn where h is a certain constant which is determined by making the co-ordinates satisfy the condition (§ 35). If 3 be an harmonic screw, then, remembering that the screws of reference are co-reci procal (§ 87), we must have n equations of which the first is (§ 102) 7, , 1 dV Pi1 ~ + fpl def- Assuming --------Ms", where M is the mass of the body, and s an unknown quantity, and developing V, we deduce the n equations 3t (4U + Mshf) + 3.,AV1 + ... + 3nAJn = 0, + (fAm + ... + 3n (Ann + Ms^uf) = 0. Eliminating dA,... 3n,we have an equation of the wth degree for s2. The n roots of this equation are all real (§ 85), and each one substituted in the set of n equations will determine, by a system of n linear equations, the ratios of the « co-ordinates of one of the harmonic screws. It is a remarkable property of the n harmonic screws that each pair of them are conjugate screws of inertia, and also conjugate screws of the potential. Let Hlt... 11,^, be n-1 of the harmonic screws, to which correspond the impulsive screws Slt... S,^. Also suppose T to be that one screw of the given screw system which is reciprocal to Slt ...S^ (§ 95), then T must form with each one of the screws If,... Hn_^ a pair of con- jugate screws of inertia (§81). But, since ... Sn_t are the screws on which wrenches are evoked by twists about If,... Hn_i respectively, it is