[ 280 ] 



XLIII. On some Extensions of Quaternions. By Sir William 

 Rowan Hamilton, LL.D., M.R.I.A., F.R.A.S., Correspond- 

 ing Member of the French Institute, Hon. or Coir. Member of 

 several other Scientific Societies in British and Foreign Coun- 

 tries, Andrews' Professor of Astronomy in the University of 

 Dublin, and Royal Astronomer of Ireland. 



[Continued from p. 51 .] 



Section V. 



[30.] TN applying to associative quines the general theory of 

 -ft- the Third Section *, we may (as has been seen) omit 

 each of the signs S v as unnecessary, the index h receiving only 

 one value in the sum thereby indicated ; and may suppress each 

 sum 2 VV as vanishing. In this manner the type IV., or the 

 formula (151), becomes, 



IV.. (egf){eff+egg)=geh.ehf; .... (191) 



while the equation (127), already derived as a sub-type from II., 

 gives, by interchanging e and/, 



^fff)(fie+fgg) = (geh){hee + hgg). . . . (192) 

 Multiplying the latter of these two equations by eff+ egg, and 

 the former by/ee +fgg, and subtracting, we eliminate the symbol 

 {egf), and find that 



^h){(ehf)(fee+fgg)-(eff+egg){hee+hgg)}=0; . (198) 

 and a similar elimination of (geh) gives the equation, 



W){W)(fie+f9ff)-Wf+esrff)(tee + hg g )}=0. . (194) 



And because (geh) = — (egh), by (34), we may make any sepa- 

 rate or combined interchanges, of e with g, and of/ with h, and 

 so vary the expression within the { }, without introducing any 

 new factor, distinct from (egf) and {egh), outside them. If, then, 

 for any particular arrangement of the four unequal indices, 

 c > f> ff> h, as chosen from among the four numbers 1, 2, 3, 4, 

 the two following conditions are not both satisfied, 



(efff)'=0, {egh)=Q, .... (195) 

 we must have, for that arrangement of the indices, a system of 

 four other equations, whereof one is 



VI.. (ehf)(fee+fgg) = (eff+egg)(hee + hgg), . (196) 



while the three others are formed from it, by the interchanges 

 just now mentioned. And if we further suppose that the two 

 sums, fee +fgg and hee + hgg, are for the same arrangement dif. 



* Comprising paragraphs [17.] to [25.], and published in the Phil. Mag. 

 for October, 1854. 6 



