306] THE GEOMETRICAL THEORY. 327 



cylindroid. For any impulsive wrench on (77, ) can be decomposed into 

 impulsive wrenches on 77 and . The first of these will generate a twist 

 velocity about a. The second will generate a twist velocity about /3. These 

 two can only compound into a twist velocity about some other screw on the 

 cylindroid (a, /3). This must, therefore, be the instantaneous screw corre 

 sponding to the original impulsive wrench on (77, ). 



It is a remarkable point about this part of our subject that, as proved 

 in 293, we can now, without any further attention to the rigid body, corre 

 late definitely each of the screws on the instantaneous cylindroid with its 

 correspondent on the impulsive cylindroid. 



We thus see how, from our knowledge of two pairs of correspondents, we 

 can construct the impulsive screw on the cylindroid (77, ) corresponding to 

 every screw on the cylindroid (a, /3). 



It has been already explained in the last article how a single known 

 pair of corresponding impulsive and instantaneous screws suffice to point 

 out a diameter of the momental ellipsoid, and also give its radius of 

 gyration. A second pair of screws will give another diameter of the 

 momental ellipsoid, and these two diameters give, by their intersection, the 

 centre of gravity. As we have an infinite number of corresponding pairs, 

 we thus get an infinite number of diameters, all, however, being parallel to 

 the principal plane of the instantaneous cylindroid. The radius of gyration 

 on each of these diameters is known. Thus we get a section 8 of the 

 momental ellipsoid, and we draw any pair of conjugate diameters in that 

 section. These diameters, as well as the radius of gyration on each of them, 

 are thus definitely fixed. 



When we had only a single pair of corresponding impulsive and instan 

 taneous screws, we could still determine one ray in the conjugate plane to 

 the diameter parallel to the instantaneous screw. Now that we have further 

 ascertained the centre of gravity, the conjugate plane to the diameter, 

 parallel to the instantaneous axis, is completely determined. Every pair of 

 corresponding impulsive and instantaneous screws will give a conjugate 

 plane to the diameter parallel to the instantaneous screw. Thus we know 

 the conjugate planes to all the diameters in the plane S. All these planes 

 must intersect, in a common ray Q, which is, of course, the conjugate 

 direction to the plane S. 



This ray Q might have been otherwise determined. Take one of the two 

 screws, of zero pitch, in the impulsive cylindroid (77, ). Then the plane, 

 through this screw and the centre of gravity, must, by Poinsot s theorem 

 already referred to, be the conjugate plane to some straight line in S. 

 Similarly, the plane through the centre of gravity and the other screw of 

 zero pitch, on the cylindroid (77, ), will also be the conjugate plane to some 



