284 Sir W. K. Hamilton on some 



respecting products of four symbols of the form (efg), 



egh.fhg.gfe.hef=ehg.fgh.gef.hfe: . . (221) 



indeed it will be found that the members of this last equation, 

 taken in their order, are respectively equal by (196) to the 

 members of the equation (205). 



With the notations l t . . .v 3) supposing that none of the twelve 

 constants Ipst vanish, and that the twelve combinations of the 

 forms »!—/«,, n-i + ii}, u l — m v r l +r 2 , are in like manner differ- 

 ent from zero, we find thus, or from the equations (199), (200), 

 combined with their consequences (207), the following among 

 other relations, in which cyclical permutation of the indices is 

 still allowed : 



? 2 ^=(w, +u 1 )(m l —u 1 ), Ms^i+^fa+^j 



p 1 s 2 ={mc,-?ic l )(m 2 -u 2 ),p ] t s ={m 3 -n 3 )(n 3 + u 3 ),,, 



-/ 2 s 2 = ( Ml + M 1 )(r l + ?- 3 ), l 3 t 3 =(u 1 —m 1 ){r ) + r^), l 



p i s 1 = [m l -n l )(n 3 +u 3 ), p l t 1 = {m l -n l )(m Q -ti 2 ). J 



The conditions (152), (153), of the fourth type, are satisfied; 

 and we have these other products, of which some have occurred 

 already, in (176), (177), in connexion with the paiticular systems 

 (A) and (B) of quines : 



s 1 Sc,s 3 =t l tJ 3 =-(r 1 + r 2 )(rs + r 3 ){r 3 + r 1 );~) 



SiP^3 = tAP3=( n i+ u \)( m i- u \){ m i- n i)'>> ' ( 224 ) 



SqSJz Ps = V3iV3 ) J 



where the two members of the equation on the last line are 

 easily proved by (223) to be respectively equal to those of (208) . 

 [33.] As yet we have only partially satisfied the conditions of 

 type II., or of the formula (123), which may for quines be 

 written thus : 



II... fgh.hff=fee.gee-fgg . gff+feh . ghe. . (225) 



Substituting for the last product in this formula its value given 

 by (211), namely 



feh.ghe=(fgg+f/ih)(gee+gff), . . . (226) 



and writing for abridgement 



v f =fee+fgg+fhh, (227) 



