A Treatise on the Theory of Screws

124 THE THEORY OF SCREWS. 1134- otherwise we should have at either intersection a twist velocity without any kinetic energy. There is no similar restriction to the axis of pitch. We thus see that O' must always lie inside the circle, but that 0 may be in any part of the plane, 135. Conjugate Screws of Inertia. We have already made much use of the conception of Conjugate Screws of Inertia. We shall here approach the subject in a manner different from that previously employed. Let a be a screw about which a rigid body is twisting with a twist velocity ä; let the body be simultaneously animated by a twist velocity ß about a screw ß. These two will compound into a twist velocity 0 about some screw 0. If the body only had the first twist velocity, its kinetic energy would be Mua2a2. If it only had the second, the energy would be Mufß2. When it has both twist velocities together, the kinetic energy is Mue2&2. Generally it will not be true that the resulting kinetic energy is equal to the sum of the components; but, under a special relation between a and ß, we can have this equality; and as shown in § 88 under these cir- cumstances a and ß are conjugate screws of inertia. The necessary condition is thus expressed :— 'ufft2 = ua2a2 + ufß2. We have now to prove the following important theorem :— Aug chord through the pole of the axis of inertia intersects the representa- tive circle in a pair of conjugate screws of inertia. For we have 02 : a2 : ß2 :: AB2 : BX2 : AX2; but if AB passes through the pole of the axis of inertia, then the centre of gravity of masses — AB2 at X, + BX2 at A, and + AX2 at B, will lie on the axis of inertia ; and, accordingly, ABhtf = BX2ua2 + ^1 X‘2uf; whence Ug20'2 = M„2d- + Uß2ß2, which proves the theorem. Or we might have proceeded thus:—From Ptolemy’s theorem (fig. 19), AB.XY = AX .BY+AY.BX-. multiplying by AB . XO', AB2. XY. XO'= AX. AB. BY. XO'+ AY. XO'. BX. AB;