A Treatise on the Theory of Screws

116 THE THEORY OF SCREWS. [128, on which would make the body commence to twist about 3, is indeterminate. Any screw in space which is reciprocal to </> would fulfil the required condition (§136). We have seen in § 96 that an impulsive wrench on any screw in space may generally be replaced by a precisely equivalent wrench upon the cylindroid which expresses the freedom. We are now going to determine the screw y, on the cylindroid of freedom, an impulsive wrench on which would make the body twist about a given screw 3 on the same cylindroid. This can be easily determined with the help of the pitch conic; for we have seen (§ 40) that a pair of reciprocal screws on the cylindroid of freedom are parallel to a pair of conjugate diameters of the pitch conic. The construction is therefore as follows:—Find the diameter A which is conjugate, with respect to the ellipse of inertia, to the diameter parallel to the given screw 3. Next find the diameter B which is con jugate, to the diameter A with respect to the pitch conic. The screw on the cylindroid parallel to the line B thus determined is the required screw y. Two concentric ellipses have one pair of common conjugate diameters. In fact, the four points of intersection form a parallelogram, to the sides of which the pair of common conjugate diameters are parallel. We can now interpret physically the common conjugate diameters of the pitch conic, and the ellipse of inertia. The two screws on the cylindroid parallel to these diameters are conjugate screws of inertia, and they are also reciprocal; they are, therefore, the principal screws of inertia, to which we have been already conducted (§ 127). If the distribution of the material of the body bear certain relations to the arrangement of the constraints, we can easily conceive that the pitch conic and the ellipse of inertia might be both similar and similarly situated. Under these exceptional circumstances it appears that every screw of the cylindroid would possess the property of a principal screw of inertia. 129. The Ellipse of the Potential. We are now to consider another ellipse, which, though possessing many useful mathematical analogies to the ellipse of inertia, is yet widely different from a physical point of view. We have introduced (§ 102) the conception of the linear magnitude va, the square of which is proportional to the work done in effecting a twist of given amplitude about a screw a. from a position of stable equilibrium under the influence of a system of forces. We now propose to consider the distribution of the parameter upon the screws of a cylindroid. It appears from §102 that if vlt v., denote the values of the quantity va for each of two conjugate screws of the potential, and if a,, a., denote the intensities of the components on the two conjugate screws of a