334] VARIOUS EXERCISES. 363 



Hence from knowing the two cylindroids eight of the co-ordinates of 

 the rigid body are uniquely fixed while the ninth remains quite indeter 

 minate. Every value for p 3 2 will give one of the family of rigid bodies for 

 which the desired condition is fulfilled. 



We have already deduced geometrically ( 306) the relations of these 

 rigid bodies. We now obtain the same results otherwise. 



The momental ellipsoid around the centre of gravity has as its 

 equation 



(x - x o y- pS +(y- 7/ ) 2 p.? + (z- ztf p s 2 - 2 (y - y ) (z - z,) I? 



- 2 (z - z ) (as - ) 4 2 - 2 (x - a? ) (y - y,} 1 3 2 - (yx, - xytf 



- ( z y - y z *f - ( xz - zx T = & 



This may be written in the form 



tf(z-ztf = R, 

 where p 3 2 does not enter into R. 



As /&amp;gt; 3 2 varies this equation represents a family of quadrics which have 

 contact along the section of R = by the plane z z^ = 0. This proves 

 that a plane through the common centre of gravity and parallel to the 

 principal plane of the cylindroid U passes through the conic along which 

 the momental ellipsoids of all the different possible bodies have contact. All 

 these quadrics touch a common cylinder along this conic. The infinite point 

 on the axis of this cylinder is the pole of the plane z - Z Q for each quadric. 

 Every chord parallel to the axis of the cylinder passes through this pole 

 and is divided harmonically by the pole and the plane z z = 0. As the 

 pole is at infinity it follows that in every quadric of the system a chord 

 parallel to the axis of the cylinder is bisected by z z . Hence a diameter 

 parallel to the axis of the cylinder is conjugate to the plane z z in every 

 one of the quadrics. Thus by a different method we arrive at the theorems 

 of 306. 



334. The Double Correspondents on Two Cylindroids. 



Referring to the remarkable homography between the impulsive screws 

 on one cylindroid V and the corresponding instantaneous screws on another 

 cylindroid U we have now another point to notice. 



If the screws on U were the impulsive screws, while those on V were the 

 instantaneous screws, there would also have been a unique homography, the 

 rigid bodies involved being generally distinct. 



But of course these homographies are in general quite different, that is to 

 say, if A be a screw in U the instantaneous cylindroid, and B be its corre 

 spondent in V the impulsive cylindroid, it will not in general be true that if 



