Balk — VonirlbulioiiH l<> Ihc Thcirji of S'l-fira, 



57 



> . 



(92) 



succeed each other simply as written round the circle ; e.g. in tlio third, the 

 suffices are 3,4, 5, 1, 2. Thus the positive terms can be written down at once. 



The terms with negative signs are obtained by omitting every alternate 

 digit on the circumference ; e.g., commencing with 3, we omit 4; take 5, omit 1 ; 

 take 2, omit 3 ; take 4, omit 5 ; take 1, and obtain 35241, the sequence in the 

 suffices in the seventh term of the expression for A. 



We can find the value of fj, in like manner, and obtain (92). 



- FAsA, Sfit/icfJ-i 



- FA3X1 Sfli,IM,fl2 



- FX1A5 SfXi/J,,IJ,j 



- FAsAi SfJh/J-i/Ji'i 

 + FAiAj SfiiiJ,2tJ-i 



+ FA3A5 Sn2fi;Hi 



+ FA5A2 (S/ij/xi/ij 



+ FXiXi SfiifXifiz 



If, indeed, we could have written down the expressions of A and fi, the 

 proof that they were the coordinates of the screw reciprocal to the five 

 given screws would have been merely a verification of the condition of 

 reciprocity; for example, using the values of fj. and A in the expression 

 S(jXi\ + /iXi), we obtain 



Sfii/jti/Xi SXsXiXs + S/iiiiifjis SXiXiXi + Sfiifii/xi SAjAaAi 



— SfJiH^fi} 8X5X2X4 — Snifiifis iSAoAjAi - Sfj.ifiif.i2 8X4X1X3 



- SXiXiXo Sfi}fitfii - 8X4X2X3 8fji4fiifxi - iSAjAoAi iSjujjujjui 



+ (SAlAlAs Sflsfllfli + 8X4X3X5 Sfl2fl4fJll + 8X4X5X2 Sfi4fXifl3, 



which vanishes identically. In like manner it can be shown that fxX is 

 reciprocal to each of the fonr other screws. We have thus obtained the 

 quaternion solution of the problem of finding that one screw which is 

 reciprocal to five given screws.* 



If the five screws belonged to a system of lower dimensions than five, the 

 screw reciprocal to them would be indeterminate. For instance, if they all 

 belonged to a 4-system, then any screw on a cylindroid reciprocal to four of 

 the screws would also be reciprocal to the fifth. We thus see that A = 0, and 

 ^ = must always be satisfied if 



(jui, Ai) ; (ju2, Aj) ; {ft,, fi,) ; (//<, Aj) ; (jui, Aj) 

 belong to a system of the fourth or any lower order. 



* This result, and certain other iluvelopments fiiot here reprodiieed), were coiumuniiated by me 

 to the Australasiaa Assouialion for tiie .^dvaneement of Science, an i iitiblishi d in tlieir Kepoit (1809), 

 p. 52. 



[8*J 



