286 Sir W. R. Hamilton on some 



tainecl by eliminating the symbol {fgh) between (229) (230), 

 and not essentially distinct from the recent equation (231) : 



VIII... hee.(fee+fhh)(gee+gff) = (hee + hff)(fee.gff-gee.fgg); (234) 

 which contains all the old and new relations, subsisting between 

 the twelve constants of the form {eff), and may be regarded as 

 an eighth type for quines. 



[34.] Denoting the first minus the second member of (234) 

 by the symbol \efgh~], we easily see that 



[efgh] =gee.(v f .hee+fggJiff) +gff(fhh.hee-hfffee) 



= hee.{v f .gee+fhh.gff)+hff.{fgg.gee-gff.fee); (235) 

 and therefore that we have, identically, 



[efgh] = lefhg]; (236) 



this last or eighth type (234) contains therefore, at most, only a 

 system of twelve equations. Interchanging / and g, and attend- 

 ing to the notation (202), we see, by (203) (234), that of the 

 three equations 



[efgh]=0, [egfh]=0, [e]=0, . . (237) 



any two include the third ; if then we only seek what new con- 

 ditions, additional to those marked (204), are to be satisfied by 

 the symbols {eff), or rather by the eight following ratios of those 

 symbols, 



eff : egg : ehh ; fee : fgg : fhh ; gee : gff: ghh ; hee : Iff: hgg, (238) 

 we need only retain at most four new equations, suitably selected 

 from among those furnished by type VIII., such as the four fol- 

 lowing, which differ among themselves by the initial letters 

 within the brackets, and so belong to different groups, 



lefgh]=0, [fghe]=0, [ghef^O, [hefg]=0; (239) 



and then to combine these with any three of the four former 

 relations (204), for example with the three first, namely 



M=0, [/]=(), [>]=0; . . . (240) 



from which the fourth equation [/;] = would follow, by means 

 of the identity, 



{ehf Q {fge) Q {gfh) Q {heg) Q ={ehg) {fgh) {gfe) {hef) Q 

 = (eff+egg)(fee+fhh){gee+ghh){kff+kgg). . . (241) 



It might seem however that the seven equations (239) and (240), 

 thus remaining, should suffice to determine seven of the eight 

 ratios (238) : whereas I have found that it is allowed to assume 

 two pairs of ratios arbitrarily, out of the four pairs (238), and 

 then to deduce the two other pairs from them. For I find that 

 it is sufficient to retain, instead of the twelve equations included 

 ' under type VIII., or the seven equations (239) (240), a system 



