Sir William Rowan Hamilton on Equations of the Fifth Degree. 349 

 k^-\^ = {k — X) (v — ex) {k — e'\) ; 

 and, secondly, by (10), 



3ic = ^ -\- ey -\- eh'', 3\ = 0" ^ ey^ -\- e^i\ 



expressions which give 



^-\ =^(e-e^)(i^-y^), 

 ,-ex = i(i-e)(^-z'), 



.-e^X = ^(e^-l)(y^-^'); 



but also, by (7), 



h^-y' = l {x" - a/") (x" - x'""), 



|3^ - 8^ = I {x" - x''') {x'" - x"), 



y--^ = :^{x"-x''){x''' -x'"); 



(65) 

 {QQ) 



(67) 



(68) 



and 



therefore, 



{e - ff) (1 — 0) (0^ - 1) = (1 - 0)^ = — 3 (0 — 0^) ; 



(69) 



r' _ \3 = _ 2-« 



3-^(e-e^){x"-j/"){af'-x^'^){x"-x'') 1 



(y- _ o;^'') (a;'" - a: (or^'' - or 0- 



J 



Thus, then, the square of the product of these six linear factors (70), and of the 

 numerical coefficients annexed, is equal to the function (64), of the twelfth di- 

 mension, which itself entered as a factor into the expression (61) for h^; and we 

 see that this square is free from the imaginary radical 0, because, by (11), 



(0 — 0^)^ = _ 3 ; 



(71) 



and that it is a symmetric function of the four roots x" , x'", x'^, x^, being pro- 

 portional to the product of the squares of their differences, as was stated in article 

 19. : so that this square (though not its root) may be expressed, in virtue of the 

 biquadratic equation (6), as a rational function of af, p, q, r; which followed 

 also from its being expressible rationally, by (64), in terms of e, v], i. 



22. Introducing now, in the expression (64), here referred to, the values 

 (16), or the relations (15), we find, after reductions : 



