58 Proceedings of the Royal Irish Academy. 



We can also obtain from (91) and (92) the conditions that must be satisfied 

 if /i, X be the coordinates of any screw of an 7i-system. For convenience, we 

 shall take the case of a o-system, and proceed as follows : — 



Being given the three screws which determine the 3-system, we take three 

 screws of the reciprocal system, and let these last be defined by the coordinates 

 (a*!. ^1)5 0*2, A3); (AI3, X3). We now take /ui, A, ; /i/j, Aj as any vectors whatever ; 

 then, as fi,\ are given by equations (91) and (92), the screw /uA must be 

 reciprocal to (/ii,A|); (/u:, A:); (^.„A3): and must therefore belong to the 

 original 3-system; and, by giving proper values to jui, A,; ^us, A5, we can 

 obtain the coordinates of any one of the screws of the system. 



Thus we are able to obtain the quaternion condition to be satisfied if six 

 screws belong to a S-system ; in other words, we are to find the condition that 

 must be satisfied by six screws so related that a body with simultaneous twist 

 velocities about these sue screws shall still be at rest. In this case, the six 

 screws must all be reciprocal to one screw. We tlierefore have merely to 

 write the condition that the sixth screw (ju. A,) shall be reciprocal to {ji. A), 

 which we have already found as the screw reciprocal to 



(/ui, A,); {jii,\i); (//I, A,); (//<, A4); (i/siAs). 



We have therefore merely to substitute for fi and X in the equation 



S(jji^ + ^,) = 0; 

 the result is accordingly ; 



- SA^AjAb . Sfiiftifii + Sfiifjiifit. oAiAjAj = 0- 



- 'SAJA4A, . Sptiftifli + Sf^^f^lfJ^. SAiAjAj 



- SAiAiAf Snifu/Js + Sfiifiifi,. SX,\i\i 



- SAiAiX«. SfiiniHi + S/iifXifi,. SA3A4A5 



- SAjAjA* . Sfiifiifii + S/jLtfiifu • SAiAjAj 

 + iSAiAtA«. SfiiHtfii - Sftifiiftt ■ SAjAjAi 

 + 5A|A|A(. Sfitfufti - SfiifAifii . SA:A,Ai 

 + SXtXiX,. Sfiifiifti - Sftiftifu- SA,AjAj 

 + SAiAiA, . Sfl,f^^fi^ - Si^ifuft,. SAiAjAs 

 + SAjA»As . Sfii/u/it - Sfit/j^i . iSAiAtX« 



This is the ^formula for the sexiant obtained in a different manner by Joly 

 (see Hamilton's Elcmcnt.% vol. ii, p. 393). It is of course the Quaternion 

 equivalent of the formula already given in Equation (41) in one form, and in 

 Treatise, pp. 37, 248 in another. 



It is worth while to note that the vector coordinates of five screws satisfy 

 certain other formulae which are, no doubt, merely properties of five pairs of 

 vectore quite unrelated. The proofs of these formulae will be derived at once 

 from the theory of screws ; but we commence by defining four vector-functions 

 A,B,C,D which are given by the following relations : — 



(93) 



