A Treatise on the Theory of Screws

95] THE PRINCIPAL SCREWS OF INERTIA. 83 93. Lemma. If a group of instantaneous screws belong to a system of the nth order, then the body being quite free the corresponding group of impulsive screws also belong to a system of the nth order; for, suppose that n + 1 twisting motions about n + 1 screws neutralise, then the corresponding n + 1 im- pulsive wrenches must equilibrate, but this would not be possible unless all the impulsive screws belonged to a screw system of the »th order. 94. Euler’s Theorem. If a free or constrained rigid body is acted upon by an impulsive wrench, the body will commence to move with a larger kinetic energy when it is permitted to select its own instantaneous screw from the screw system P defining the freedom, than it would have acquired, had it been arbitrarily restricted to any other screw of the system. Let Q be the reciprocal system of the (6 — n)th order, and let P' be the screw system of the nth order, consisting of those impulsive screws which, if the body were free, would correspond to the screws of P as instantaneous screws. Let i) be any screw on which the body receives an impulsive wrench. De- compose this wrench into components on a system of six screws consisting of any n screws from P', and any 6 — n screws from Q. The latter are neutralised by the reactions of the constraints, and may be omitted, while the former com- pound into one wrench on a screw £ belonging to P'; we may therefore replace the given wrench by a wrench on £ If the body were perfectly free, an impulsive wrench on £ must make the body twist about some screw a on P. In the present case, although the body is not perfectly free, yet it is free so far as twisting about a is concerned, and we may therefore, with reference to this particular impulse about £ consider the body as being perfectly free. It follows from § 92 that there would be a loss of energy if the body were compelled to twist about any screw other than a, which is the one it naturally chooses. 95. Co-ordinates in a Screw System. The co-ordinates of a screw belonging to a specified screw-system can be greatly simplified by taking n co-reciprocal screws belonging to the given screw system as a portion of the six screws of reference. The remaining 6 - n screws of reference must then belong to the reciprocal screw system. It follows that out of the six co-ordinates a/, ... «6 of a screw a, which belongs to the system, 6 — n are actually zero. Thus we are enabled to give the more general definition of screw co-ordiuates which is now enunciated. If a wrench, of which the intensity is one unit on a screw a, which belongs to 6—2