438 Proceedings of the Royal Irish Academy. 



reference the six common screws of the two systems, then we have at 

 once for the co-ordinates of the screw corresponding to a — 



(ll)ai, (22)03, (33)a3, (44) a^, {5 b) a,, (66) ««. 



"When these substitutions are made in the determinants it is obvious 

 that they still yanish ; we hence have the important result that 



The screws corresponding homographically to the screivs 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 correspon- 

 dents. We thus see that while two reciprocal screw systems of the n*^ 

 and (6 - 7i'^) orders respectively have as correspondents systems of 

 the same orders, yet that their connexion as reciprocals is divorced by 

 the homographic transformation. 



Reciprocity is not therefore an invariantive attribute of screws or 

 screw systems. There are, however, certain functions of eight screws 

 analogous to anharmonic ratios which are invariants. These functions 

 are of considerable interest, and they are not without physical signifi- 

 cance. 



"We have ohQ&dij {Screws, p. 163) discussed the important function 

 of six screws which is caUed the Sextant. This function is most 

 concisely written as the determinant (ai /So y^ 84 €5 te) where a, /3, y, 

 8, e, I, are the screws. In Sylvester's language we may speak of 

 the six screws as being in i^ivolution 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 bfr 

 easv to show that functions of the following form are invariants : — 



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 

 cliaiiged into its corresponding screw. 



