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



are divisible, respectively, by the three products 



{x^ — x^) (Xa — x^), {x.^ — Xi) (Xe — Xa), (x^ — x^) (x^ — Xj) ; (128) 



and, therefore, the factor (126) is divisible by the product 



(x^ — x,)(x^ — x,), (129) 



the quotient of this division being a rational and integral and homogeneous func- 

 tion of the five roots x, v?hich is no higher than the third dimension, and which 

 it is not difficult to calculate. 



31. In this manner we are led to establish an equation of the form : 



V345- V354 + ^'(V453- V543) +^(V534- V435) = (1 " ^) K" ^3) i^^-^^) ^l' (130) 



in which if we make 



2N, = 10N', + (5-f)N% (131) 



we have 



(^2-*3)(^4-«5) (^2 - -^3) (^4 - ^5) 



Effecting the calculations indicated by these last formulae, we find 



n', = |(m".-m',), N".= -fM"„ (133) 



m', and m", being determined by the equations (99) and (100) ; and, therefore, 

 with the meaning (98) of m„ we find the relation : 



n,= -125m,. (134) 



Thus, the first of the three factors (119) may be put under the form : 



— 125(1 -e)(x,-x,) {x,-x,)m,; (135) 



■:.■:■ - ) 

 in deducing which, it is to be observed, that the first term, Xa* x^, of the formula 



(107) for Waicdc gives, by (108), the five following terms of Xjcd«: 



5Xa* Xi + SiCj' Xa + 5Xe' Xa + 5x/ X^ + 5Xe* Xa l ( 1 36) 



and these five terms of x give, respectively, by (111), the five following parts 

 of Y<afa: 



