326 THE THEORY OF SCREWS. [305, 



Let G be this intersection, and draw OP parallel to a and equal to the 

 radius of gyration about GP, which we have shown to be known from the 

 fact that a and ij are known. Let X be the plane conjugate to GP in the 

 moraental ellipsoid, then this plane is also known. 



In like manner, draw GQ parallel to ft and equal to the radius of 

 gyration about GQ. Let Y be the plane, conjugate to GQ, in the momental 

 ellipsoid. 



Let PI and P 2 be the perpendiculars from P, upon X and Y respec 

 tively. 



Let Q l and Q 2 be the perpendiculars from Q, upon X and Y respec 

 tively. 



Then, from the properties of the ellipsoid, it is easily shown that 



PI ^2 = Qi Qv 



This is the second geometrical relation between the two pairs of screws 

 a, ?; and ft, . Subject to these two geometrical conditions or to the two 

 formulse to which they are equivalent the two pairs of screws might be chosen 

 arbitrarily. 



As these two relations exist, it is evident that the knowledge of a second 

 pair of corresponding impulsive screws and instantaneous screws cannot 

 bring five independent data as did the first pair. The second pair can bring 

 no more than three. From our knowledge of the two pairs of screws together 

 we thus obtain no more than eight data. We are consequently short by 

 one of the number requisite for the complete specification of the rigid body 

 in its abstract form. 



It follows that there must be a singly infinite number of rigid bodies, 

 every one of which will fulfil the necessary conditions with reference to the 

 two pairs of screws. For every one of those bodies a is the instantaneous 

 screw about which twisting motion would be produced by an impulsive 

 wrench on 77. For every one of those bodies ft is the instantaneous screw 

 about which twisting motion would be produced by an impulsive wrench 

 on f. 



306. A System of Rigid Bodies. 



We have now to study the geometrical relations of the particular system 

 of rigid bodies in singly infinite variety which stand to the four screws in the 

 relation just specified. 



Draw the cylindroid (a, ft) which passes through the two screws a and ft. 

 Draw also the cylindroid (q, ) which passes through the two corresponding 

 impulsive screws 17 and It is easily seen that every screw on the first of 

 these cylindroids if regarded as an instantaneous screw, with respect to the 

 same rigid body, will have its corresponding impulsive screw on the second 





