124 THE THEORY OF SCREWS. [134- 



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 must always lie inside the circle, but that 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 d ; let the body be simultaneously animated by a twist velocity /? 

 about a screw /3. These two will compound into a twist velocity about 

 some screw 6. If the body only had the first twist velocity, its kinetic 

 energy would be Mv-ffi. If it only had the second, the energy would be 

 Muf fP. When it has both twist velocities together, the kinetic energy is 

 Mufti*. 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 /8, we can have this equality ; and as shown in 88 under these cir 

 cumstances a and /3 are conjugate screws of inertia. The necessary condition 

 is thus expressed : 



ufu* = u a 2 d* + M/j 2 /3 2 . 



We have now to prove the following important theorem : 



Any 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 



0- tct 2 :^ 2 :: AR- : BX* : AX 2 ; 



but if AB passes through the pole of the axis of inertia, then the centre of 

 gravity of masses AIF at X, + BX&quot; at A, and + AX- at B, will lie on the 

 axis of inertia ; and, accordingly, 



uf = BX-u a 2 + 

 whence 



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 , 



