6all — Contributions to the Throrij of Screws. 49 



With this substitution, 



- >S'(,x,A, + Aa/x.) = T\^ T\, j {p, + ih) cos - r/ sin 0| . (65) 



Hence we have the quaternion proof that in all cases 



i I (Pi + V'^ coa9-d sin i (66) 



is the virtual coefficient of two vector-screws of pitches jh, Pi, and distance d, 

 where is the right-handed angle between them* 



The condition that two screws (^iXi), (/^jAj) shall be reciprocal is now very 

 simply expressed by stating that their virtual coellicient vanishes or 



S{,m\, + nX) = 0. (67) 



We have from this the quaternion proof of the well-known property thus 

 stated. 



If a screw {n\) be reciprocal to two screws (juiA,) and (juaAs), it is reciprocal 

 to every screw on the cylindroid which passes through (juiAi) and (jujAs) . 



For, if hi and A-j be any two scalars, we may represent the typical screw on 

 the cylindroid by 



[kiHi + hjx^, (/o,Ai + /:jA=) ; 

 and if 



<S(/iXi +iui^) = 0, and .S(,(A2 + /iA) = 0, 

 then 



S\fi{l:iXi + A-jAj) + \{kiiii + hfi2)\ = 0. 



More generally, we have the following theorem : — 



If a screw {n, A) be reciprocal to each of the n screws 

 (fi,, Ai) ; {fi2, Aj) ; . . . (/^„, A„), 

 it will then be reciprocal to aU screws of the group 



[kifli + h_Hi . . . knfln), (JCiXi + AvAa . . . + InK), 



whatever kik, . . . may be. 



We may enunciate the same principle in a still more general manner which 

 includes the whole theory of reciprocal screw systems as follows : — 



If each of the m screws 



(yui. A.i), (^2, A-) . . . (/i,„, A,„) 

 is reciprocal to all of the n screws 



then all screws of the type 



{k,ni + A-2JU2 . . . + k„fi,„), (kiXi + k-iX. . . . + k,„X^) 

 will be reciprocal to all possible screws of the type 



{kifMi + k-zfl.' . . . + k,n'lXm), (A'/Ai' + A-/A,' . . . + km\„,'], 



whatever may be the values of the scalars 



fcl, Ki . . . A"„„ A"i , A"2 ... A"m . 



* See Joly, in Hamilton's " Elements," vol. ii., p. 391. 



[7*] 



