PAPERS READ IN SECTION A. 



4- 



1.— ON THE QUATERNION EXPRESSION FOR THE CO-ORDINATES 

 OF A SCREW RECIPROCAL TO FIVE GIVEN SCREWS. 



By Sir ROBERT BALL, F.B.S., Lowndeun Professor of Astronomy and Geometry, Unioersily 



of Cambridge. 



One of the most fundamental principles in the theory of screws 

 is that which asserts that, if five screws are not contained in a system 

 of order less than five, then one screw, but only one, can be found 

 which is reciprocal to each of the five screws. We shall now prove 

 this by the quaternion representation, and we shall obtain the 

 co-ordinates of the reciprocal screw in terms of the co-ordinates of 

 the given screws. 



Let the co-ordinates of four screws be respectively — 



/*! K ; i^'.K; Ms '^j ; /*4 \ ; 



and let x^, x„, x^, x^ be any four scalars. Then the 4-system will be 

 found by giving all values to x^ x^ x^ x^ in the co-ordinates — 



a^i ft, -l-^,/A,-f oTj/x^ + ar^yu.^, x)^,-\-x^„-\-x^^ + x^^ (i.) 



Por if |Li A be the co-ordinates of a screw reciprocal to each of the four 

 screws, then (see Joly's Manual of Quaternions, p, 204) — 



S(A/x. + /xX,) = 0; S(Aft -t-/AA.,)=0; S(A/x,-|-/A3) = 0; S(Aft^-|-/AAj=0, 

 and cojisequently — 



S{A(^,/>i,+^,/x,-4-a:3/Aj + a%/xJ -!-/x(a:.A, ■\- x„K-{- x^^-Y x ^y^—i). (ii.) ; 

 so that /A, A is a screw reciprocal to every screw of the type (i.) It is 

 thus shown that all screws which are reciprocal to the original four 

 screws are also reciprocal to every screw of the type (i.), and hence 

 the latter must be the general representation of the screw of a 

 4-system. 



It is well known that if A,, A,, A^, A^ be any four vectors whatever, 

 A,SA,A3A,-A,SA3A,A,-)-A3SA,A,A,-A,SA,A,A3=0 (iii.) ; 

 let us make in (ii.) — 



y=:SA„A,A, ; ir,= — SA.A.A • a,' =SA,AA •, .i;,= — SA.A.A ,; 

 then from (ii.) and (iii.) we have 



SA(/x,SA,AjA^ — yiijSAjA^A, + /XjSA^AjA, — /^^8AjA,A,)=0. 

 The interpretation of this equation is that every screw /jt, A which is 

 reciprocal to the four given screws must also be at right angles to the 

 vector — 



ftjSAAjA^ — ftjSA^A^A, + /^,*^\'^,'\ — 1*-^^)^^^ (iv-) 



We know that the locus of screws which are reciprocal to four giA^en 

 screws is a cylindroid, and hence we obtain the following theorem : — 



Given four screios fi, A, ; yu.. A,; /a,A, ; /x^A^ ; then the axis of tlie 

 cylindroid reciprocal to these four screws is parallel to the rector. 



/XjSAjA^A^ — /x^SAjA^A, -{-/AjSA^AjA, — /x^tSAjAA^ 



