A Treatise on the Theory of Screws

129] FREEDOM OF THE SECOND ORDER. 117 wrench of unit intensity on a screw a, which also lies upon the cylindroid, then— Va2 = + if**2- From the centre of the cylindroid draw two straight lines parallel to the pair of conjugate screws of the potential, and with these lines as axes of x and y construct the ellipse, of which the equation is— Vi2x2 + v.ty2 = H, where II is any constant. If r be the radius vector in this ellipse, we have— - = a, and - — a2; r r whence by substitution we deduce— which proves the following theorem:— The linear parameter va on any screw of the cylindroid is inversely pro- portional to the parallel diameter of a certain ellipse, and a pair of conjugate screws of the potential are parallel to a pair ol conjugate diameters ut the same ellipse. This ellipse may be called the ellipse of the potential. The major and minor axes of the ellipse of the potential are parallel to screws upon the cylindroid, which, for a twist of given amplitude, correspond to a maximum and minimum potential energy. When the body has to relinquish its original position of equilibrium by the addition of a wrench on a screw y to the forces previously in operation, the twist by which the body may proceed to its new position of equilibrium is about a screw 0, which can be constructed by the ellipse of the potential. Determine the screw </> (on the cylindroid of freedom) which is reciprocal to y (§ 26), then </>, and the required screw 0, are parallel to a pair of conjugate diameters of the ellipse of the potential. The common conjugate diameters of the pitch conic, and the ellipse of the potential, are parallel to the two screws on the cylindroid, which we have designated the principal screws of the potential (§ 101). When a body is disturbed from its position of equilibrium by a small wrench upon a principal screw of the potential, then the body could be move to the new position of equilibrium required in its altered circumstances by a small twist about the same screw.