A Treatise on the Theory of Screws

216 THE THEORY OF SCREWS. [211 to the generators of a right circular cone. All the screws reciprocal to y form a cylindroid, and is the one screw of the system which is parallel to the nodal line of the cylindroid. The virtual coefficient of and r/ is greater than that of 7/ with any other screw. If 6 be a screw about which, when a body is twisting with a given twist velocity it has a given kinetic energy, then we must have + u.202 + u202 - E w + 0/ + 0.2) = 0, where E is a constant proportional to the energy. It follows that the locus of 0 must be a conic passing through the four points of intersection of the two conics uffi + u22022 + ut20s2 = 0, 0^ + 02 + 0.2 = 0. ihe four points in which these two conics intersect correspond to the screws about which the body can twist with indefinite kinetic energy. These four points A, B, C,D being known, the kinetic energy appropriate to every point P can be readily ascertained. It is only necessary to measure the anharmonic ratio subtended by P, at A, B, C, D, and to set off' on a straight line distances u^, uf, u32, h~, so that the anharmonic ratio of the four points shall be equal to that subtended by P. This will determine h2, which is proportional to the kinetic energy due to the unit twist velocity about the screw corresponding to P. A quiescent rigid body of mass M receives an impulsive wrench of given intensity on a given screw 77; we investigate the locus of the screw 0 belonging to the three-system, such that if the body be constrained to twist about 0, it shall acquire a given kinetic energy. It follows at once (§ 91) that we must have E(u 202 + u2022 + u32032) = (p101Vi + p.ß.2p2 + p-iØ^, where E is proportional to the kinetic energy. The required locus is there- fore a conic having double contact with the conic of inertia. It is easy to prove from this that E will be a maximum if : piVi = u202 : p.^.2 = u3203 : päVä; whence again we have Euler’s well-known theorem that if the body be allowed to select the screw about which it will twist, the kinetic energy acquired will be larger than when the body is constrained to a screw other than that which it naturally chooses (§ 94). Å somewhat curious result arises when we seek the interpretation of a tangent to the conic of infinite pitch. This tangent must, like any other straight line, correspond to a cylindroid; and since it is the polar of the