Ball — Contributions to the Thcorij of Screws. 55 



For if be the origin, and 0' the point indicated by the vector a, then the 

 coordinates of the screw with respect to the origin 0' are {(/x + V\a), X.). 

 Hence the coordinates of the reflected screw with regard to ff are 

 - i{n + V\a) i, iXi. We have now to transfer these coordinates back to the 

 original origin ; and we have 



fi' = - i [ji + V\a) i - ViKia = - i/ni ~ i{ VXa) i - ViXia 



= - ifxi - ViXai - ViXia = - i/xi + 2 Sai V\i 

 X' = iXi. 

 It is easy to verify this by showing that the reflected screw again reflected 

 reverts to the original screw. 



VI. — Quaternion Investifjatio7i of the Screw reciprocal to Jive given Screivs. 



If Hi, Xi ; fii, Aa ; JU3, A3 ; fii, Ai be the coordinates of four screws, and 

 if Xi, Xi, X3, Xi be any four scalars, then the four-system defined by the four 

 screws will consist of all screws with the coordinates 



(Xifjii + x^Ui + Xifii + Xifii) ; (oJiXi + x^X^ + x^X^ + «iAi), 



where a.',, x,, x^, a'j have all possible values. 



Let fi, X be any screw on the cyliudroid reciprocal to the four-system ; 

 then, since this must be reciprocal to every individual screw, we must have 

 for all scalar values of Xi, x^, x^, Xi and for every screw {fi, \) on the reciprocal 

 cylindroid 



S\X {xifii + Xifx^i + x^fXi + Xifi^ + (I (a-'iX, -t- it'jAj + a'sAj + roiAi)] = 0. (86) 

 As this is to be true for all values of Xi, x^, Xz, Xt, it will be true if 



Xi — 0A2A3A4 J X2 = — OA,3A4Al J .1*3 = /SA1A1X2 ', Xl = — 0A1A2A3. 



It is, however, a well-known formula in Quaternions that if A,, Ao, X3, \ be 

 any four vectors, then 



X,SX.XX, - X,SX,XX + X,SX,XX - X,SXXX, = 0. (87) 



Accordingly the equation (86) becomes 



SX (JU1/SA2A3A4 - /liSXiXiXi + fiiSXiXiXz — fXiSXiXiXs) = 0. 

 Thus we prove that each generator of the reciprocal cylindroid must be at 

 right angles to the vector 



fxiSXiXsXi — /xaSAaAjAi + fiiSXiXiX-, - fuSXiXiXs, (88) 



and consequently this vector must be parallel to the axis of the cylindroid 

 reciprocal to the four screws (juiAi), {1x2X2), (jusAj), (JU1A4). 



If (p, A) be also reciprocal to a fifth screw fi^, A5, then it must be at right 

 angles, not only to the vector (88), but also to any similar vector (89) obtained 

 by taking any other combination of four screws out of the five, for example, 

 ^(2>SA3A4A5 - lizSXiXiXz + fiiSXiXiXa - fuSXiXiXi. (89) 



B. I. A. PROC, VOL. XXVIII., SECT. A. [8] 



