A Treatise on the Theory of Screws

106'] HARMONIC SCREWS. 99 106. Discussion of the Results. For the position of the body before its displacement to have been one of stable equilibrium, it is manifest that the co-ordinates must not increase indefinitely with the time, and therefore all the values of s2 must be essen- tially positive, since otherwise the values of 0/, ... 9n' would contain expo- nential terms. The 2n arbitrary constants are to be determined by the initial circum- stances. The initial displacement is to be resolved into n twists about the n screws of reference (§ 95). This will provide n equations, by making t = 0, and substituting for $/,••• 9,', in the equations just mentioned, the amplitudes of the initial twists. The initial twisting motion is also to be resolved into twisting motions about the n screws of reference. The twist velocities of these components will be the values of ,1 ... when t = 0 ; whence at dt we have n more equations to complete the determination of the arbitrary constants. If the initial circumstances be such that the constants H,, Hn are all zero, then the equations assume a simple form: 0i = fnH1 sin (sji + c), On =finHy sin (sd + c). The interpretation of this result is very remarkable. We see that the co-ordinates of the body are always proportional to hence the body can always be brought from the initial position to the position at any time by twisting it about that screw, whose co-ordinates are proportional to /n> but, as we have already pointed out, the screw thus defined is a harmonic screw, and hence we have the following theorem:— If a rigid body occupy a position of stable equilibrium under the influence of a conservative system of forces, then n harmonic screws can be selected from the screw system of the nth order, which defines the freedom of the body, and if the body be displaced from its position of equilibrium by a twist about a harmonic screw, and if it also receive any small initial twist velocity about the same screw, then the body will continue to perform twist oscillations about that harmonic screw, and the amplitude of the twist will be always equal to the arc of a certain circular pendulum, which has an appropriate length, and was appropriately started. The integrals in their general form prove the following theorem:— A rigid body is slightly displaced by a twist from a position of stable equilibrium under the influence of a system of forces, and the body receives 7—2