334 THE THEORY OF SCREWS. [309, 



It is shown ( 137) that the points which represent the instantaneous 

 screws, and the points which represent the corresponding impulsive screws, 

 form two homographic systems. A well-known geometrical principle asserts 

 ( 146), that if each point on a circle be joined to its homographic corre 

 spondent, the chord will envelop a conic which has double contact with the 

 circle. It is easily seen that, in the present case, the conic must be the 

 hyperbola which touches the circle at the ends of the diameter XY, and 

 has the rays AP and BQ for its asymptotes. The hyperbola is completely 

 defined by these conditions, so that the pairs of correspondents are uniquely 

 determined. 



Every tangent, 1ST, to this hyperbola will cut the circle in two points.. 

 I and S, such that 8 is the point corresponding to the impulsive screw, and 

 / is the point which marks out the instantaneous screw. We thus obtain 

 a concise geometrical theory of the connexion between the pairs of cor 

 responding impulsive screws and instantaneous screws on a cylindroid which 

 contains two of the principal screws of inertia of a free rigid body. 



For completeness, it may be necessary to solve the same problem Avhen 

 the cylindroid is defined by two principal screws of inertia lying along the 

 same principal axis of the rigid body. It is easy to see that if, on the 

 principal axis, whose radius of gyration was a, there lay any instantaneous 

 screw whose pitch was p a , then the corresponding impulsive screw would 

 be also on the same axis, and its pitch would be p n where p n x p a = a?. 



310. Analogous Problem in a Three-system. 



Let us now take the case of the system of screws of the third order, 

 which contains three of the six principal screws of inertia of a free rigid 

 body. 



Any impulsive wrench, which acts on a screw of a system of the third 

 order, can be decomposed into wrenches on any three screws of that system, 

 and consequently, on the three principal screws of inertia, which in the 

 present case the three-system has been made to contain. Each of these 

 component wrenches will, from the property of a principal screw of inertia, 

 generate an initial twist velocity of motion around the same screw. The 

 three twist velocities, thence arising, can be compounded into a single twist 

 velocity about some other screw of the system. We desire to obtain the 

 geometrical relation between each such resulting instantaneous screw and 

 the corresponding impulsive screw. 



It has been explained in Chap. XV. how the several points in a plane 

 are in correspondence with the several screws which constitute a system 

 of the third order. It was further shown, that if by the imposition of 

 geometrical constraints, the freedom of a rigid body was limited to twisting 



