288 Sir W. R. Hamilton on some 



which may be formed from (243) (244) by cyclical permutation 



f off. hee-ehh . hff=gff. (ehg) ;l 

 , n J »« • ¥f+ e 9U • ?iee=gee . {ehg) ; \ 

 U} " ' ] -('•/, • eff+hgg.ehh)=ghh . (ehg) ; \ ' ***> 



Multiplying then the equations (243) and (244) by {ehg) , and 

 observing that the identity (220) gives 



(ehg) .(g¥)o=(^e + kff)(ehf) , . . (248) 



we find, on substitution of the first for the second members of 

 (247), that the results are divisible by hee + hff; and that thus 

 the elimination of the third pair of ratios (238), between (243) 

 (244) (247), or between the four equations (242), conducts to 

 expressions of the recent forms, namely, 



hee . egg—ehh . hgg=fgg . {ehf) ;"\ 



v e . hgg + eff . hee =fee . (ehf) ; 1 



-(v h .egg + hff.ehh)=fhh.{ehf) Q) [ * ^ 



Vh.eff—v e . hff = v f . (ehf) . - 



A similar analysis may be applied to effect the elimination of the 

 fourth pair of ratios (238), with results entirely analogous. On 

 the whole then it is found, that the four equations (242) express 

 such connexions between the four pairs of ratios (238), as to 

 satisfy not only the two remaining equations, (245) and (246), 

 of their own groups, (e) and (/), but also the six other equations 

 of the two other groups, (g) and (A), included under type VIII.; 

 namely 



I*)*... O#]=0, [gfeh]=0, [ghef]-- 

 (h) .... [hefg] =0, \hfeg] =0, [hgef] : 



for the first line is satisfied by the ratios (249), and the second 

 line by the analogous ratios, which are found in a similar way. 

 Thus all the twelve equations of type VIII. are satisfied, if we 

 satisfy only four suitably selected equations of that type ; for 

 example, the equations (242) : which was what we proposed to 

 demonstrate. 



[36.] The eighty equations of association, assigned in the 

 Third Section, between the twenty-four constants l x . . . u s , or 

 ( e fg)> i e ff)> have therefore, by the recent analysis, been ultimately 

 reduced to sixteen ; namely the four equations which thus remain 

 from the last type VIII. ; and the twelve others which were con- 

 tained in the type III., established in that earlier Section : and 

 which (as was lately remarked) leave still no fewer than eight 



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