On some Extensions of Quaternions. 281 



ferent from zero, and write for abridgment, as a definition, 



(ehf) =(fee+fgg)- l (eff+egg)(hee + hgg), . (197) 



the four equations furnished by the formula (196), which may 

 be regarded as a sixth type for quines, maybe concisely expressed 

 as follows : 



(ehf) = (ehf) Q , ( 9 hf)=.{ghf) 0) -\ 



(efh) = (efh) Q) (gfh) = (gfh) .S' ' V } 



With the notations l x . . u 3 , for the symbols {efg), (eff), we find 

 thus that unless I x and p x both vanish, we must have the four 

 equations, 



^(«i-'"i) = K- ?? 2 )( ? 2 + ? '3) ; s z {n x -m x ) = (m 3 -n 3 )(r 2 + r 3 ) ;\ , lQQ , 



Psi r 9 + r sl = K + "a) [ n i ~ m \ ) I Ps( r s + r 3 ) = {™ 3 ~ w s) ( re i - m i) H 

 and that unless s x = 0, f, = 0, then 



— 4K-?n 2 ) = ( ra i + M i) (% + %); ^( M 2-%) = ( ? 'i + ? '2)( w 3 + M 3)n (200) 



^( W 3 + M 3) = ( W l- m l)( M 2- m 2 ); S 2( W 3 + M3) = (»'l+?-3)( M 2- m 2)-^ 



[31.] Supposing then that no one of the twelve symbols (efg) 

 vanishes, and that each of the twelve sums eff+ egg is also dif- 

 ferent from zero, the various arrangements of the four indices 

 efgh give us a system of twenty-four equations, included in the 

 new type VI., or in any one of the four formulae (198); which 

 equations may, by (34), be arranged in twelve pairs, as follows : 



(ehf) = (ehf) =-(hef) (201) 



It might seem that twelve equations between the twelve symbols 

 of the form (eff) should thus arise, by the comparison of two 

 expressions for each of the twelve symbols of the form (efg) ; 

 but if we write for abridgement 



l9] = (fee+fgg)(fhh+fgg){(ehf) 0+ (hef >0 } ) . (202) 



and observe that by the definition (197) of the symbol (ehf) , 

 we have then 



0] = (<#+ egg) (fhh +fgg) (hee + hgg) 



+ Wf+h99){fee+fgg)(ehh + egg), . . (203) 



we shall see that this quantity Qgr] is independent of the arrange- 

 ment of the three indices e, f h; and therefore that the twelve 

 equations between the twelve symbols (eff), obtained through 

 (201), reduce themselves to the, four following relations, 



[ e ]=0, [f]=0, [g]=0, [*]_0j . . (204) 

 which are not even all distinct among themselves, since any 

 three of them include the fourth. An easy combination of the 

 two first or of the two last of these four relations (204) conducts 



