Ball — Contributions to the Theory of Screws. 63 



Multiplying the llrst by VXAi, VX^X^, and FAiAj respectively, and taking 

 the scalar, SX\0<i = '16' A 1X2X3, 



SXAaAi = .'(^SAiXaXa, 



<SXXiX-> = a'3iSX 1X2X3 ; 



whence 



SXX2X3 1SX1XA3 <SXiX2X 



6'XiXsX3 (81X1X2X3 <SiAiX2X3 



But the expression on the right hand is a linear vector function of X of the 

 most general type.* If we denote it by ^X, we have 



H = ^\. (101) 



Another proof of this important theorem may be noted as follows : — 



If /iX be a screw reciprocal to the three screws (yuiX,), (jUjXa), (jUaXj), we have 



S{n\x + VO = 0, 8{fi\^ + Aju,) = 0, S(//Aj + A/^3) = 0. 

 But, by a fundamental quaternion formula, 



^SXiX^X, = FA2A3 . SXmx + FA3A, . SX,n + FAiAs . SX^in. 

 Whence from the three equations of reciprocity just written, we have 



/uSA.AaXa = rX,X,8^x^X + rXiX^Sfi^X + VX2X,Sfi,X, (102) 



again showing that ju is a linear vector function of A. 



Being given any linear vector function ^, then by taking different vectors A, 

 the pair of coordinates (^A, X) will trace out the screws of the 3-system 

 corresponding to (p. This theorem is due to Joly,t and it is a discovery of 

 much importance in the theory, inasmuch as it shows the perfect correspon- 

 dence between the 3-system and the linear vector function. 



When A is given, then /a = (A) is known ; and thus we see that in a 

 3-system there is always one screw parallel to any given direction. The 

 pitch of the screw is S^X . A"', and the perpendicular from the origin on 

 the screw is V(j)X . A''. The equation of any screw of the 3-systeni is 



p = V^X . A'' + X X. 



Jolyf has also shown that if (^A, A) represents a 3-system, then (- ^'A, A) 

 represents the reciprocal 3-system, where as usual ^'A is the function 

 conjugate to <^A. This beautiful theorem shows the intimate connexion 

 between the theory of reciprocal screw systems of the third order and 

 the properties of the linear vector function. 



In conclusion I add a few illustrations to show how the Theory of Screws 

 responds to treatment by the methods of Quaternions. I shall assume that 



* See Report of the British Association for the Advancement of Soifince. Dublin, I90S, p. 611. 

 t Hamilton's " Elements," vol. ii., Appendix, p. 391. 

 tliiil., p. 392. 



B.I. A. PROC, vol.. XXVIIl., SKCT. A. [9] 



