323] THE GEOMETRICAL THEORY. 353 



If we regard T and G 1 G 2 as two screws of zero pitch, and draw the 

 cylindroid through these two screws, then any impulsive wrench about a 

 screw on this cylindroid will have the same instantaneous screw for either 

 of the two bodies to which it is applied. 



For such a wrench may be decomposed into forces on T and on G 1 G 2 ] 

 these will produce, in either body, a twist about a, and a translation parallel 

 to a, respectively. We therefore obtain the following theorem : 



If two rigid bodies have different centres of gravity, G { and 6r 2 , and if 

 their radii of gyration about the ray G^ z are equal, there is then a 

 cylindroid of screws such that an impulsive wrench on any one of these 

 screws will make either of the rigid bodies begin to twist about the same 

 screw, and the instantaneous screws which correspond to the several screws 

 on this cylindroid, all lie on the same ray Gfi^, but with infinitely varied 

 pitch. 



It is to be remarked that under no other circumstances can any im 

 pulsive screw, except the ray GiG z , with zero pitch, have the same instan 

 taneous screw for each of the two bodies, so long as their centres of gravity 

 are distinct. 



We might have demonstrated the theorem, above given, from the results 

 of 303. We have there shown that, when an impulsive screw and the 

 corresponding instantaneous screw are given, the rigid body must fulfil five 

 conditions, the nature of which is fully explained. If we take two bodies 

 which comply with these conditions, it appears that the ray through their 

 centre of gravity is parallel to the instantaneous screw, and we also find 

 that their radii of gyration must be equal about the straight line through 

 their centres of gravity. 



If two rigid bodies have the same centre of gravity, then, of course, any 

 ray through this point will be the seat of an impulsive wrench on a screw of 

 zero pitch such that it generates a twist velocity on a screw of infinite pitch, 

 parallel to the impulsive screw. This will be the case to whichever of the 

 two bodies the force be applied. We have therefore a system of the third 

 order (much specialized no doubt) of impulsive screws, each of which has the 

 same instantaneous screw for each of the two bodies. In general there will 

 be no other pairs of common impulsive and instantaneous screws beyond 

 those indicated. 



Under certain circumstances, however, there will be other screws possess 

 ing the same relation. 



We may suppose the two momental ellipsoids to be drawn about the 

 common centre of gravity. These ellipsoids will, by a well-known property, 

 possess one triad of common conjugate diameters. In general, of course, the 

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