A Treatise on the Theory of Screws

138 THE THEORY OF SCREWS. [148- Any ray through P, the intersection of the axis of inertia with the tangent at H, cuts the circle in two points, A' and A", either of which will receive the same kinetic energy from the given impulse. 149. Euler’s Theorem. If the body be permitted to select the screw about which it will commence to twist, then, as already mentioned, § 94, Euler’s theorem states that the body will commence to move with a greater kinetic energy than if it be restricted to some other screw. By drawing the tangent from P (not, however, shown in the figure) we obtain the point of contact B, where it is obvious that the ratio of the perpendiculars on PH and PQ is a maximum, and, consequently, the kinetic energy is greatest. It follows from Euler’s theorem that B will be the instantaneous screw corresponding to A as the impulsive screw. The line BH is the polar of P, and, consequently, BH must contain O', the pole of the axis of inertia. We are thus again led to the construction (§ 140) for the instantaneous screw B; that is, draw AOH, and then HO'B. 150. To determine a Screw that will acquire a given Twist Velocity under a given Impulse. The impulsive sci’ew being given, and the intensity of the impulsive wrench being one unit, the acquired twist velocity (§ 147) will vary as (Fig. 30), AH A’Q ■ If, therefore, the twist velocity be given, this ratio is given. A' must then lie on a given ellipse, with H as the focus and the axis of inertia as the directrix. This ellipse will intersect the circle in four points, any one of which gives a screw which fulfils the condition proposed in the problem. The relation between the intensity of the impulsive wrench and the twist velocity generated can be also investigated as follows: Let P, Q, R, S be points on the circle (Fig. 31) corresponding to four im- pulsive screws, and let P', Q', R', S' be the four corresponding instantaneous screws deduced by the construction already given. Let p, q, r, s denote the intensities of the impulsive wrenches on P, Q, R, S, which will give the units of twist velocity on P', Q', R', S'. Supposing that impulsive wrenches on P, Q, R neutralize, then the corresponding twist velocities generated on P', Q’, R' must neutralize also. In the former case, the intensities must be proportional to the sides of the triangle PQR; in the latter, the twist