Extensions of Quaternions. 287 



fa £?MT TC ati0DS 0f the tyP 6 J^t mentioned; namely two 

 pans, selected from any two of the four groups, which have (for 

 each group and also for each pair) a common initial letter within 

 the brackets ; for instance, these two pairs of equations : 



[ef ff h-]=0, [egfh1=0. [fegh^O, [fgeh]=0; . (242) 



t W W - e ?r £ maUy ,f ? (jU arUtrar y constants (for example 



h IrS^t ff **&-*% hee, hff, hgg, from which 'all 



ouines It Tr d) " tke remltin V S y stem ofassoeiative 



SSbJS T?A ° f th V™tigaticm by which this important 



ro" 1 T1 ^ CCted i may be P rese *ted in the following way. 



Paisofrattn, X T^ 8 (242) C ° Unect the thr « ^ 



pans ot latios (238), in such a manner that when any two of 



those three pairs are assumed, or known, the third can be deter- 



S . UC . e, e Wlt 1 ^ the 1 mterpretati0n ( 197 ) of the symbol 

 {9hf) , we easily find that those two equations (242) give, 



■ f gee . hgg -ghh . hee = (ghf) .fee;") 

 (e) ..J v r . hee +gff. hgg = (gkf) .fgg; I . (24 3) 

 L ~ (®* • gee + hff.ghh) = (ghf) .fkli;] 

 because we find that/,,, fgg,fhh are proportional to the left- 

 hand members of these last equations (243) ; and that the sum 

 of he two first of those left-hand members is identically equS 

 two fi^'oAt ^gfmee^hgg). For thesame reason^ 

 two first of these three equations (243) express really only one 

 relation name y that which is contained in the second equaS 

 (242), although they do so under different forms, both of which 

 it is useful to know; and it is convenient to have ready also tins 



togetVeT ' ] ^ addmS thethreee ^ atlon « (243) 



( e )-- ^.Vff-v g .hff=v f .{ghf)^ . . (244) 



which, like those equations (243) themselves, we shall consider 



two of T "fi t0 thC gI '° Up {C) > bGCaUSe the ^ are ^ derived from 

 VIIT 2?A r - e ^ Uatl0nS ° f that 8^'oup, included under t™ 

 VIII which m the recent notation [efghl have e for the r 

 ini ,a letter; and because the third eqmWon of tha tgit p 

 included under the same type, namely ° P ' 



(•).. W9~]=0, (245) 



S^h^^iT tHen V fc y the elimination of the symbol 

 GjA/)o between the first and third of the equations (243) In 



lSn tW °!r , CqUatl ° DS ( ? 42) indudc a *** <>' the 

 same type VIII., and bclongmg to the same group (/), namely 



(/)■• [/%]=0; ( 2 46j 



because they conduct to the following system of expressions, 



