2GG THE THEORY OF SCREWS. [249- 



Whcri these substitutions are made in the determinants it is plain that 

 they still vanish ; we hence have the important result that 



The screws corresponding homographically to the screws of an n-system 

 form another n-system. 



Thus to the screws on a cylindroid will correspond the screws on a 

 cylindroid. It is, however, important to notice that two reciprocal screws 

 have not in general two reciprocal screws for their correspondents. We thus 

 see that while two reciprocal screw systems of the nth and (6 ?i)th orders 

 respectively have as correspondents systems of the same orders, yet that 

 their connexion as reciprocals is divorced by the homographic transforma 

 tion. 



Reciprocity is not, therefore, an invariantive attribute of screws or screw 

 systems. There are, however, certain functions of eight screws analogous to 

 anharmoriic ratios which are invariants. These functions are of considerable 

 interest, and they are not without physical significance. 



250. Analogy to Anharmonic Ratio. 



We have already ( 230) discussed the important function of six screws 

 which is called the Sexiant. This function is most concisely written as the 

 determinant (a^^Js^s^s) where a, /3, 7, B, e, are the screws. In Sylvester s 

 language we may speak of the six screws as being in involution when their 

 sexiant vanishes. Under these circumstances six wrenches on the six screws 

 can equilibrate ; the six screws all belong to a 5-system, and they possess one 

 common reciprocal. In the case of eight screws we may use a very concise 

 notation; thus 12 will denote the sexiant of the six screws obtained by 

 leaving out screws 1 and 2. It will now be easy to show that functions of the 

 following form are invariants, i.e. the same in both systems: 



12 . 34 



13. 24 



It is in the first place obvious that as the co-ordinates of each screw enter to 

 the same degree in the numerator and the denominator, no embarrassment 

 can arise from the arbitrary common factor with which the six co-ordinates of 

 each screw may be affected. In the second place it is plain that if we replace 

 each of the co-ordinates by those of the corresponding screw, the function 

 will still remain unaltered, as all the factors (11), (22), &c., will divide out. We 

 thus see that the function just written will be absolutely unaltered when 

 each screw is changed into its corresponding screw. 



By the aid of these invariant functions it is easy, when seven pairs of 

 screws are given, to construct the screw corresponding to any given eighth 



