344 THE THEORY OF SCREWS. [317, 



Draw pairs of corresponding planes through X and X . The locus of 

 their intersection will be a quadric S&quot;, which also contains the four double 

 points. 



S&quot; and C, being of the second and the third order respectively, will 

 intersect in six points. Two of these are on X and X , and are thus dis 

 tinguished. The four remaining intersections will be the required double 

 points, and thus the problem has been solved. 



These double points correspond to the principal screws of inertia, which 

 are accordingly determined. 



In the case of freedom of the fifth order, the geometrical analogies which 

 have hitherto sufficed are not available. We have to fall back on the 

 general fact that the impulsive screws and the corresponding instantaneous 

 screws form two homographic systems. There are five double screws be 

 longing to this homography. These are the principal screws of inertia. 



318. Correlation of Two Systems of the Third Order. 



It being given that a certain system of screws of the third order, P, is 

 the locus of impulsive screws corresponding to another given system of the 

 third order, A, as instantaneous screws, it is required to correlate the corre 

 sponding pairs on the two systems. 



We have already had frequent occasion to use the result demonstrated 

 in 293, namely, that when two impulsive and instantaneous cylindroids 

 were known, we could arrange the several screws in corresponding pairs 

 without any further information as to the rigid body. We have now to 

 demonstrate that when we are given an impulsive system of the third 

 order, and the corresponding instantaneous system, there can also be a 

 similar adjustment of the corresponding pairs. 



It has first to be shown, that the proposed problem is a definite one. 

 The data before us are sufficient to discriminate the several pairs of screws, 

 that is to say, the data are sufficient to point out in one system the corre 

 spondent to any specified screw in the other system. We have also to 

 show that there is no ambiguity in the solution. There is only one rigid 

 body ( 293) which will comply with the condition, and it is not possible that 

 there could be more than one arrangement of corresponding pairs. 



Let a, /3, 7 be three instantaneous screws from A, and let their corre 

 sponding impulsive screws be 77, , in P. In the choice of a screw from 

 a system of the third order there are two disposable quantities, so that, 

 in the selection of three correspondents in P, to three given screws in A, 

 there would be, in general, six disposable coordinates. But the fact that 

 a, ?; and @, g are two pairs of correspondents necessitates, as we know, 



