Ball — Contributions to the Theorij of Screws. 67 



function, we have in general by hhe known properties of aeif- conjugate 

 linear vector-functions 



{<i> - p) { V{^ - p] X (0 - p) ,^ I = m,,V\n. (110) 



where nip is a constant, so far as X and n are concerned, depending only 

 on 2^, 'T-nd t^® coefficients of the latent cubic appropriate to <p. The actual 

 value of m.„ is thus found. Multiply (110) by any vector v and take the 

 scalar. Then the first side of (110) becomes 



s{'i> - p)i vii, -p)x (<i.-p)fi]v = s(^-p) v\ Vi<p -p)\ i<p-p)i^\ 



= S(i,-p)\.(i,-p),i.(i,-2})v. 

 The second side of (110) becomes when the scalar is taken 



mpSVXfi.v = nipSXuv. 

 Thus we have 



_ S{<p -2} )X.{<t>'P)n.(<p~p)v _ 8{i, - p) i (0 -p )j{i>-p)Jc 



because of the very remarkable property of linear vector functions which 

 affirms that itip is unchanged whatever three vectors be chosen as X/n'. 

 Substituting for (107) and (108) in (110) we have 



{<t> - p) VVpX . Vpiii = - vip VXfi ; 

 but WpX .Vpfi = - pSXftp, 



and consequently (^ - p)pSXfip = mp VX/i, 



whence multiplying by p and taking the scalar 



Sp{,^ - p]p = mp. (112) 



It is indeed astonishing to find that so concise a formula as this should 

 contain the theory of the 3-system of screws. 



Much further development no doubt awaits the investigation of the 

 relations of the Theory of Screws to Quaternions. 



