54 



PROCEEDINGS OP SECTION A. 



The order of the suffices in the several terms of X will be seen 

 as follows : — 



Writing; the digits 1, 2, 3, 4, o round 

 the circumference of a circle consecu- 

 tively, we notice that in each of the five 

 positive terms of A the five dibits succeed 

 each other simply as written round the 

 circle — e.^., in the third the suffices are 

 3, i, 5, 1, 2. Thus the positive terms 

 can be written down at once. 



The five terms with nej^ative signs 

 each have a row of digits, which are 

 also obtained from the succession on 

 the circle. In this case, however, we 



are to omit every alternate digit on the circumference — e.q., com- 

 mencing with 3 we omit i, take 5, omit 1, take 2, omit 3, take 4, 

 omit 5, take 1, and obtain 3, 5, 2, 4<, 1— and this gives the sequence 

 of suffices in the seventh term of (vii.) 



We have thus found A one <»F the two vectors (/>t, A) which are 

 required for the determination ot the screw reciprocal to five given 

 screws, and it now only remains to find ihe other vector fjL. 



If in the equation (ii.) we had taken — 



^, ^ S/x, /Xj /A^ ; X„'= — S/x./x^//,, ; :rj=S/x^/x, /x, ; ^^4= — S/t,/x,/Xj ; 

 we see that /x must be at right angles to — 



A,S/Xjyu,./x^ — AjS/x^/ix^/x, -\- AjS/x^ft,/Xj — A^|Uj|(Xj/x^ = ; 



and, following the same reasoning as in the case of A, we obtain for /x 

 the value — 



/x= — VAjA.S/Xj^a^/Xj^ 



VAjAjS/X^/A; /Xj 



— VAjAS/Xj /Xj /^,j 



V A^A, S^j ylX, /X^ 



-YA.A,8/x,/..3/x, Uviii) 



-|- A'^AjA.S/x, /x^/x, 

 + YA,A,8/x^/x,/Xj 

 + A^A,A_|'^/Xj yUj /x, 



+ VA^A, S/X^ /X, JUL, J 



It is easy to verify tbat these values of A and /x denote a screw 

 reciprocal to each of ihe five given screws. Obviously, SA/x/,- -j- SXk fx. 

 = tor all integral values of k 5. For example— 



SA/x^= S/x^)[x, /XjSAjA^Aj -|- S/x^/Xj/XjSA A. A, + S/x^/x^ ju,,SAjAjA^ 



— I^/x^/x, /XjSAjAjA^ — b/x^/Xj/XjISAjA^A, — IS/x^/Aj/XjSA^AjAj 



SA^/x= — JSA^A_A,8/Xj/i,^/Xj — yA^AjA^S/x^/x. /x, — ISA^A. A S/x, ft jx^ 



-\- SA^A AjS/Xj /x, fjL^ + kSA^AjA. kS/Xj /x^ /x, + SA^Aj.A,N/x^ /x, /x., 



whence 



S A/x^ -|- S A^ ^ ^ 0. 

 We have thus obtained the complete quaternion solution of the 

 fundamental problem of finding the vector co-ordinates of a screw 

 reciprocal to five given screws. 



