QUATERNIOK EXPRESSION FOR SCREWS, ETC. 



53 



I£ fx, A be reciprocal to the five screws — 



IX,. X,; fx,. X, ; ^3, X3 : fx^, X^ ; /x., X^ ; 



then it must be at right angles not only to A^ector (iv.) but also to 

 any similar vector obtained by taking any other combination of four 

 screws out of the five — e.r/., 



fx,SX^X^X^ — yu,jS/\^A,/\„ -j- ju-^SAjAjXj — /AjSAjXsXi (v.) 



Hence A must be parallel to the vector part of the product of these 

 two vectors (iv.) and (v.), for which we have the expression — 



V/x,/x,(SA,A,A,SA,AA.) 1 



+ Vja,/./SA.A>,8A^A.A, - SA^A/jSA/.A,) 

 + \>,/x^(SA,A,A„8A,A;A3 — SA.XAjSA^AjAJ 

 + V^,^5(SAA3A^SA;aA3) 

 -|- V^/Aj/i.,(SA,A3A^SA,A3A^) 



+ V/x,/x3(SA",A3A^SaXX-) 

 + V/.3/.,(SA,A3A^S\A,A,) 

 + V/oi./^,(SA,A,A^SA3A,AJ 

 4- V^' /x^ (SA3A. A^S A. A, A3 — S A, A, A3SA3A. AJ 



Kvi.) 



+ ^f^j'^X'^KKK^KKK) 



J 



At present this expression does not seem symmetrical, though it 

 is obvious that the vector parallel to the screw reciprocal to five screws 

 must be symmetrical in regard to the^^e five sci'ews. We can give the 

 expression in the desired form by the following Lemma : — 



If a, /3, y, 8, and be any five vectors, then — 



this can easily be verified by substituting for a and /3 from the 

 equations — 



a = Ay + B8+C^; y8= A'y+ B'8 + C'^ ; 



when A, B, C, A'. B', C' are scalars. Thus we find that — 



SA3A/, . SA/jA, - SA3\A,SA/.A,= SA^AjA, SA,A,A, 

 SA^A,A, SA,A;A,-SA;a„A3SA>,A, = SA,A3A^ 8AA,A, 

 SA,A, A, SaXa' — SA A,A,SaX/V = SA„A A • SA A A, 



Introducing these values into (vi.) and dividing out by the common 

 factor SA^AjA^ (we assume that no three of the five given screws are 

 complanar), we obtain for A, the vector parallel to the screw reciprocal 

 to the five given screws, the symmetrical expression — 



X=+Yix,ix,SX^X^X^^ 

 + A'^/Xj /x.SA^A.A, 



+ Y IX^fx]SXA\X: 



+ Yfji^fi,SXXX] 



+ Yix.ix,'SX,xX I 

 — Yfx^/x^SX.X^ I 



— V|U3/X.SA\A, 



-Yfx^fx]Hx]x\ 

 — V/x^/x,SA,AjA„J 



(vii.) 



