A Treatise on the Theory of Screws

88] THE PRINCIPAL SCREWS OF INERTIA. 79 on A> will make the body commence to twist about A,. Hence A1 and also A2 and A3 are principal screws of inertia. We shall now show that with the exception of the n screws here deter- mined, generally no other screw possesses the property. Suppose another screw S were to possess this property. Decompose the wrench on *S' into n wrenches of intensities S/', ... Sn" on Alt ... An’, this must be possible, because if the body is to be capable of twisting about »S' this screw must belong to the system specified by -4i, ... An. The n impulsive wrenches on -41, ... An will produce twisting motions about the same screws, but these twisting motions are to compound into a twisting motion on & It follows that the component twist velocities ... Sn must be proportional to the intensities S", ... Sn". But if this were the case, then every screw of the system would be a principal screw of inertia; for let X be any impulsive screw of the system, and suppose that Y is the corresponding instantaneous screw, the components of X on Alt ... An, have intensities X", ... Xn', these will generate twist velocities equal to y y // cy // 1 , • • • Q “>/ H > >□1 and these quantities must equal the components of the twist velocity about Y. But the ratios Ä Ån s»’ S? are all equal, and hence the twist velocities of the components on the screws of reference of the twisting motion about Y must be proportional to the intensities of the components on the same screws of reference of the wrench on X. Remembering that twisting motions and wrenches are compounded by the same rules, it follows that Y and X must be identical. As it is not generally true that all the screws of the system defining the freedom possess the property enjoyed by a principal screw of inertia, it follows that the number of principal screws of inertia must be generally equal to the order of the freedom. 88. Kinetic Energy. The twisting motion of a rigid body with freedom of the «th order may be completely specified by the twist velocities of the components of the twisting motion on any n screws of the system defining the freedom. If the screws of reference be a set of conjugate screws of inertia, the expression for the kinetic energy of the body consists of n square terms. This will now be proved. If a free or constrained rigid body be at rest in a position L, and if the