A Treatise on the Theory of Screws

90 THE THEORY OF SCREWS. [100- Introducing these expressions we find, for the condition that 9 and £ should be reciprocal, ^1 (-^•11^’1 4" • • • Aln<f>n ) 4" • • • + 9n (^Anl<f)1 + ... 4* ~ 0. This may be written in the form :— ÄA'</>l' + ••• Alm9n'<l>f + A12 (0/</>./ + </>/) + ... =0. But this equation is symmetrical with respect to 9 and </>, and therefore we should have been led to the same result by expressing the condition that </> was reciprocal to y. When 9 and </> possess this property, they are said to be conjugate screws of the potential, and the condition that they should be so related, expressed in terms of their co-ordinates, is obtained by omitting the accents from the last equation. If a screw </> be reciprocal to then $ is a conjugate screw of the potential to 9. If we consider the screw 9 to be given, we may regard the screw system of the fifth order, which embraces all the screws reciprocal to y. as in a certain sense the locus of f. All the screws conjugate to 9, and which, at the same time, belong to the screw system G by which the freedom of the body is defined, must constitute in themselves a screw system of the (»— l)th order. For, besides fulfilling the 6 — n conditions which define the screw system C, they must also fulfil the condition of being reciprocal to y; but all the screws reciprocal to 7 — n screws constitute a screw system of the (ft — l)th order (§ 72). The reader will be careful to observe the distinction between two conju- gate screws of inertia (§ 81), and two conjugate screws of the potential. Though these pairs possess some useful analogies, yet it should be borne in mind that the former are purely intrinsic to the rigid body, inasmuch as they only depend on the distribution of its material, while the latter involve extrinsic considerations, arising from the forces to which the body is submitted. 101. Principal Screws of the Potential. We now prove that in general n screws can be found such that when the body is displaced by a twist about any one of these screws, a reduced wrench is evoked on the same screw. The screws which possess this property are called the principal screws of the potential. Let a be a principal screw of the potential, then we must have, § 99:— ' + 2^ daj ’ and («— 1) similar equations.