350 THE THEORY OF SCREWS. [320- 



cau be no further ambiguity. The correspondent in A to every other screw 

 in P is completely known. To show this it is only necessary to take two 

 pairs from A and P and the pair just found. We have then three corre 

 sponding pairs. It has been shown in 307 how the correspondence is 

 completely determined in this case. 



Of course the fact that 8 may be any screw on a cylindroid is connected 

 with the fact that in this case the rigid body had one indeterminate element. 

 For each of the possible rigid bodies & would occupy a different position on 

 its cylindroidal locus. 



321. Systems of the Fifth Order. 



It remains to consider the case where two screw systems of the fifth 

 order are given, it being known that one of them P is the locus of the 

 impulsive screws which correspond to the several screws of the other system 

 A regarded as instantaneous screws. 



Let P be the screw reciprocal to P, and A the screw reciprocal to A. 

 Then from the theorem of 319 it follows that an impulsive wrench on A 

 would make the body commence to move by twisting about P . We thus 

 know five of the coordinates of the rigid body. There remain four inde 

 terminate elements. 



Hence we see that, when the only data are the two systems P and A , 

 there is a fourfold infinity in the choice of the rigid body. There are conse 

 quently four arbitrary elements in designing the correspondence between the 

 several pairs of screws in the two systems. 



We may choose any two screws 77, , on P, and assume as their two corre 

 spondents in A any two arbitrary screws a and ft, provided of course that 

 the three pairs A , B , rj, a, and , ft fulfil the six necessary geometrical 

 conditions ( 304). Two of these conditions are obviously already satisfied by 

 the circumstance that A and P are the reciprocals to the systems A and P. 

 This leaves four conditions to be fulfilled in the choice of a. and ft. As each 

 of these belongs to a system of the fifth order there will be four coordinates 

 required for its complete specification. Therefore there will be eight disposable 

 quantities in the choice of o and ft. Four of these will be utilized in making 

 them fulfil the geometrical conditions, so that four others may be arbitrarily 

 selected. When these are chosen we have four coordinates of the rigid 

 body which, added to the five data provided by A and P , completely define 

 the rigid body. 



322. Summary. 



We may state the results of this discussion in the following manner : 

 If we are given two systems of the first, or the second, or the third order 



