QUATERNION EXPRESSION FOR ><CREWS, ETC. 



55 



Let Wj, M„ w„ w_, oj. be five vectors parallel to the axes of the cylin- 

 droids obtained by leading out in succession each of the five screws 

 and then forming the cylindroid reciprocal to the remaining four — 



(o,=yu.3SA^A.A, — |x^i^X.X^X^-lr fx^i^X^X^X^ — yu,,SA A^A. j 

 fi>j=:/x,^SAA.A, — yw,.SA,A,A4 + /x,8A,A^A. — jn^SA^A A, \- (ix.) 

 w^=^..SA,A,A, — />(,,SAjA,A. -h/AjSA^A-A, — yu^SAA A, | 

 o^=//.,SA,AjA_^ — /"■I'^'^AAi "t"/"-','^\^A: — /A^SA.AAj j 

 Substituting these quantities in the values for A in (vii.), we obtain — 



If we interchange A and /* in all the terms of (ix.), and represent by 

 /;,, . . 7/. what OJ,, . . w res|)eetivcly would then become, we have — 

 2fM= V(X-q^ + A,ry, + A,7?3 + A^t;^ + A.,/.) . 



A useful verification of these formulas is provided by the follow- 

 ing considerations. It has been arranged that «ith respect to a given, 

 origin O, the vector co-ordinates of a certain screw are p., X. Suppose 

 that the origin be now transferred to another point O'. so that the 

 vector 00' = p we require to find the altered co-ordinates of the 

 screw. It is plain ihar,as the direction of the screw is independent of 

 the origin, there need be no change in A. To find the (;hange in fx, we 



o 



proceed as follows, Pig. 2 : — Let PP' he the si-rew and OP and O'P' 

 the perpendiculars on the screw from O and O', then we have — 



OP' = V - + .r A = p -f V ; 

 A A 



and, as of course the expressions for the pitch of the screw must be 

 the same in both cases, 



A A 



adding these two equations, we have — 



U. ji.' 



"^ -^ xX = p+ . 

 XX 



operating with both sides upon A and equating the vectors — 



/i.' = ^ — V p A, 

 we thus learn that — 



If (/A, A) be the vector co-ordinates of a screw with respect to a 

 ■certain origin O, then the vector co-ordinates of the same screw with 

 respect to an origin O' will be {(/<.— V p A), A} where p is the vector 



