323] THE GEOMETRICAL THEORY. 351 



of corresponding impulsive screws and instantaneous screws, all the corre 

 sponding pairs are determined. There is no arbitrary element in the 

 correspondence. There is no possible rigid body which would give any 

 different correspondence. 



If we are given two systems of the fourth order of corresponding 

 impulsive screws and instantaneous screws then the essential geometrical 

 conditions ( 281), not here making any restriction necessary, we can select 

 one pair of correspondents arbitrarily in the two systems, and find one rigid 

 body to fulfil the requirements. 



If we are given two systems of the fifth order of corresponding impulsive 

 screws and instantaneous screws then subject to the observance of the geo 

 metrical conditions we can select two pairs of correspondents arbitrarily in 

 the two systems, and find one rigid body to fulfil the requirements. 



If we are given two systems of the sixth order of corresponding impul 

 sive screws and instantaneous screws then subject to the observance of 

 the geometrical conditions we can select three pairs of correspondents 

 arbitrarily in the two systems, and find one rigid body to fulfil the 

 requirements. 



The last paragraph is, of course, only a different way of stating the results 

 of 307. 



323. Two Rigid Bodies. 



We shall now examine the circumstances under which pairs of impulsive 

 and instantaneous screws are common to two, or more, rigid bodies. The 

 problem before us may, perhaps, be most clearly stated as follows : 



Let there be two rigid bodies, M and M . If M be struck by an impulsive 

 wrench on a screw 6, it will commence to twist about some screw X. If M 

 had been struck by an impulsive wrench on the same screw 6, the body would 

 have commenced to twist about some screw /u,, which would of course be 

 generally different from \. If 6 be supposed to occupy different positions 

 in space (the bodies remaining unaltered), so will \ and p move into corre 

 spondingly various positions. It is proposed to inquire whether, under 

 any circumstances, 6 could be so placed that X and //, should coincide. In 

 other words, whether both of the bodies, M and M , when struck with an 

 impulsive wrench on 6, will respond by twisting about the same instantaneous 

 screw. 



It is obvious, that there is at least one position in which 6 fulfils the 

 required condition. Let 6^ G&amp;lt;&amp;gt; be the centres of gravity of M and M . Then 

 a force along the ray G l 6r 2 , if applied either to M or to M , will do no more 

 than produce a linear velocity of translation parallel thereto. Hence it 



