348 THE THEORY OF SCREWS. [319, 



Let P be the cylindroid which is composed of the screws reciprocal to P. 

 Let P l} P 2 , P 3 , P 4 be any four impulsive screws on P. Let A 1} A 2 , A 3 , A 4 

 be the four corresponding instantaneous screws on A. 



Take any screw a. on the cylindroid P . Let 77 be the corresponding 

 impulsive screw. Since a is reciprocal to all the screws on P it must be 

 reciprocal to Pj. It follows from the fundamental property of conjugate 

 screws of inertia, that 77 must be reciprocal to A^. In like manner we can 

 show that 77 is reciprocal to A 2 , A z , and A t . It follows that 77 is reciprocal 

 to the whole system A, and therefore must be contained in the reciprocal 

 cylindroid A. Hence we obtain the following remarkable result, which is 

 obviously generally true, though our proof has been enunciated for the 

 system of the fourth order only. 



Let P and A be any two systems of screws of the nth order, and P and A 

 their respective reciprocal systems of the (ft~n)th order. If P be the collec 

 tion of impulsive screws corresponding severally to the screws of A as the 

 instantaneous screws for a certain free rigid body ; then, for the same free 

 rigid body A will be the collection of impulsive screws which correspond to 

 the screws of P as instantaneous screws. 



320. Systems of the Fourth Order. 



Thus we see that when we are given two systems of the fourth order P 

 and A as correspondingly impulsive and instantaneous, we can immediately 

 infer that, for the same rigid body, the screws on the cylindroid A are 

 the impulsive screws corresponding to the instantaneous screws on the 

 cylindroid P . 



We can now make use of that instructive theorem ( 293) which declares 

 that when two given cylindroids are known to stand to each other in this 

 peculiar relation, we are then able, without any further information, to mark 

 out on the cylindroids the corresponding pairs of screws. We can then 

 determine the centre of gravity of the rigid body on which the impulsive 

 wrenches act. We can find a triad of conjugate diameters of the momental 

 ellipsoid, and the radii of gyration about two of those diameters. Hence 

 we have the following result : 



If it be given that a certain system of the fourth order is the locus of the 

 impulsive screws corresponding to the instantaneous screws on another given 

 system of the fourth order, the body being quite unconstrained, we can then 

 determine the centre of gravity of the body, we can draw a triad of the 

 conjugate diameters of its momental ellipsoid, and we can find ike radii of 

 gyration about two of those diameters. 



There is still one undetermined element in the rigid body, namely, the 



