A Treatise on the Theory of Screws

118 THE THEORY OF SCREWS. [130- 130. Harmonic Screws. The common conjugate diameters of the ellipse of inertia, and the ellipse of the potential, are parallel to the two harmonic screws on the cylindroid (§ 104). This is evident, because the pair of screws thus determined are conjugate screws both of inertia and of the potential. If the body be displaced by a twist about one of the harmonic screws, and be then abandoned to the influence of the forces, the body will continue to perform twist oscillations about that screw. It the ellipse of inertia, and the ellipse of the potential, be similar, and similarly situated, it follows that every screw on the cylindroid will be a harmonic screw. 131. Exceptional Case. We have now to consider the modifications which the results we have arrived at undergo when the cylindroid becomes illusory in the case con- sidered (§ 120). Suppose that £ and £ were a pair of conjugate screws of inertia on the straight line about which the body was free to rotate and slide independently. Then taking the six absolute principal screws of inertia as screws of reference, we must have (§ 97)— v , / , P( dR\ / dR\ „ spr Hi + F bi + f- , =0, \ dp J \ dpj where p denotes the screw of zero pitch on the same straight line. Expanding this equation, and reducing, we find + 4 + dp) + 16 I2 = °- This result can be much simplified. By introducing the condition that as in § 120— = (p, + %)2 + (% + p4y + (Vs + Vliy, we obtain (//b /(] /?\ 2 =8- Hence we can prove (§ 133) that in this case the product of the pitches of two conjugate screws of inertia is equal to minus the square of the radius of gyration about the common axis of the screws.