282 Sir W. R. Hamilton on some 



to this other formula, which is equivalent to three distinct equa- 

 tions : 



(eff+ ehh) (fee +fgg) (ghh +gee) (hgg + hff) = 



(eff+ egg) (fee +fhh) (ghh +gff) (hgg + hee) j . (205) 

 and which may also be thus written, 



(^f) (egf) =(ehf) (gef) . . . . (206) 



With the notations /, . . u 3 , the twenty-four equations (201) are 

 sufficiently represented by the formula? (199) and (200), if cyclical 

 permutation of the indices be employed; the four equations 

 (204) take the forms, 



('•i + r^(n 3 + u 3 )(m 2 -n 2 ) = (»•, + r 3 )(mo-u Q )(m 3 -n 3 ), 1 



(»j fMj) («a + "a) («a + Kg) = (»h - «i) ('» 2 ~ "2) ( m 3 ~ w a) > * 



whereof the equation on the second line may be obtained from 

 the product of the three represented by the first line : and the 

 three equations (205) or (206) are included in the following, 

 which is evidently a consequence of (207), 



(r l + r 3 )(n i +u J )(n 2 + u^)(n 3 -m 3 ) = (r l +r q )(u l -m 1 )(n 2 -mc l )(u 3 ~m 3 ). 



[32.] As regards the quotients and products of the symbols 

 (efg), which we shall continue to write occasionally without 

 parentheses, we have by type VI., or by (197) (198), 



e]£ ejf+egg 



ffV 9ft+9 e e 



ehf.efh=(eff+egg)(ehh + egg); . . . (210) 



ehf.gfh = (eff+egg)(ghh+gee); . . . (211) 



eliminating (ehf) between the two last of which three equations, 

 we obtain a relation of the same form as the first. Interchanging 

 g and h in (210), and subtracting, we find that 



I.. ehf.efh-egf.efg=(egg) a —(chh)*;. . (212) 



but this is precisely what the type I., or the formula (113), be- 

 comes for quines, when we cyclically advance the four indices in 

 the order fegh; the conditions (117) (118) of that first type will 

 therefore be satisfied, if we satisfy those of the sixth. Had we 

 divided instead of subtracting, we should bave found 



ehf.efh _ e/J + eg g , 



egf.efg-eff+ehk l i0) 



To interchange /, g, and divide, would only lead by (210) to 

 another equation of this last form ; but the same operations per- 

 formed on (211) conduct to the equation 



ehf _ ghh + gee . 



heg~fhh+fee' {X) 



