A Treatise on the Theory of Screws

78 THE THEORY OF SCREWS. [86- Hence we find that for all screws on the cylindroid represented by 02, 0, 0 the energy will vary as the pitch when the twist velocity remains the same. It appears from the representation of the Dynamical problem in chap. XII. that in this case all the screws of the cylindroid 03, 02, 0, 0 must be principal screws of Inertia. The number of principal screws of inertia is therefore infinite in this case. (See Routh’s Theorem, Appendix, Note 2.) 87. Enumeration of Constants. It is the object of this article to show that there are sufficient constants available to permit us to select from the screw system of the ??th order expressing the freedom of a rigid body, one group of n screws, of which every pair are both conjugate and reciprocal, and that these constitute the principal screws of inertia (§ 78). To prove this, it is sufficient to show that when half the available con- stants have been disposed of in making the n screws conjugate (§ 81) the other half admit of adjustment so as to make the screws also co-reciprocal. Choose At reciprocal to Blt ... B6_n, with n — 1 arbitrary quantities; A., reciprocal to Alt Blt ... Btl^n, with n — 2 arbitrary quantities, and so on, then the total number of arbitrary quantities in the choice of n co-reciprocal screws from a system of the nth order is , „ , n(n — 1) n — 1 + 7i — 2 ... + 1 = v 7. Hence, by suitable disposition of the n(n — 1) constants it might be anticipated that we can find at least one group of n screws which are both conjugate and co-reciprocal. We have now to show that these screws would be the principal screws of inertia (§ '18). We shall state the argument for the freedom of the third order, the argument for any other order being precisely similar. Let Ai, A2, A3 be the three conjugate and co-reciproeal screws which can be selected from a system of the third order. Let Blt B.., Bs be any three screws belonging to the reciprocal screw system. Let J?,, R,, R3 be any three impulsive screws corresponding respectively to Alt A2, A3 as instantaneous screws. An impulsive wrench on any screw belonging to the screw system of the 4th order defined by 2?1; B2, B. will make the body twist about A, (§ 82), but the screws of such a system are reciprocal to A2 and for since A, and A2 are conjugate, Ri must be reciprocal to A2 (§ 81), and also to since A3 and A. are conjugate. It follows from this that an impulsive wrench on any screw reciprocal to _42 and will make the body commence to twist about A), but A, is itself reciprocal to A2 and j43, and hence an impulsive wrench