A Treatise on the Theory of Screws

144 THE THEORY OF SCREWS. [156— For, suppose the body receives a small initial displacement about X, this will evoke a wrench on H, found by drawing XO"Y and YOH (§ 155). But the effect of a wrench on H will be to produce twist velocity about a screw found by drawing HOY and YO'X, i.e. X itself (§ 140). Hence the wrench evoked can only make the body still continue to twist about X, and harmonic vibration on X will be the result. Similar reasoning, of course, applies to Y. 157. Small Oscillations in general. The initial displacement, and the initial twist velocity of the body, can always be decomposed into their respective components on X and Y. The resulting small oscillations can thus be produced by compounding simple harmonic twist oscillations about X and K If it should happen that 0' and 0" become coincident, then every screw would be a harmonic screw. If 0 and O’ coincided, then every screw would be a principal screw of inertia (§ 86). If 0 and 0" coincided, then every screw would be a principal screw of potential. 158. Conclusion. The object proposed in this Chapter has now been completed. It has been demonstrated that the representative circle affords a concise method of exhibiting many problems in the Dynamics of a Rigid System with two degrees of freedom, so long as the body remains near its initial position. The geometrical interest of the enquiry is found mainly to depend on the completely general nature of the constraints. If the constraints be specialized to those with which mechanical problems have made us familiar, it will