372 Sir William Rowan Hamilton on Equations of the Fifth Degree. H 



'^''V = Hi9 + '/H20 + a/h,, 4- \/h,, + Vh 



bM 

 III 2 



•19 

 ^19 



Vh„ 



24' 



= H,9 + \/h2o + 0Vh„ + 1/H22 + 0\/h,3 — /h24 : 



(a'") 



and 



• R45 ^^ ^*1< 



* 435 "19 



H„ 



\/h2o + VH^TyXa + Vh23 + Vh24 ; 

 /h^o + e^^^i — /h,2 + 0'Vhj3 + v'h24; 

 \/h2„ + eVHo, — 7h^+ <?\/h,3+ a/h^^; 



(b'") 



together with twelve other expressions similar to these, and to those already 

 marked (a) and (b), but involving the functions v' and v" ; we shall have, as the 

 same author has remarked, a system of converse formulas, analogous to (c) and 

 (d), for the determination of the values of the eighteen quantities h,, ... H24. 

 Among these, we shall content ourselves with here examining one of the most 

 simple, namely the following : 



H,« = i (v"'3./ + y"'J + v'".34^ + y'\.J + V"J + v"',3/) ; (220) 



for the purpose of showing, by an example, that this quantity is not Independent 

 of the arrangement of the five roots of the original equation of the fifth degree. 

 45. Resuming with this view the equation (147), and the arrangement of 

 the roots (148), we find the following system of the twenty-four values of the 

 function Xjcie : 



-- 500; X3,,, = - 90; x,,,3 = 240; x^,, = 500 ; 



^4235 • 



^2453= 1165; 



^2634 9" 5 ^5243 V^^ J 



— — 935 ; X5324 — — 515 ; x^^^ — — 1 165 ; 



'■3425 



= 515; 



'•4352 



= -620; X3^,= -295; x^^=145; 



*4532 ' 



= — 240 ; 

 70; 



Xj543 — b20 ; X5234 — — 720 ; X4325 — 720 ; X3452 — — 70 ; 



= — 145; X4253 = 375; 



= -375; x^, = 295; 



which give, by (211), 



4y"'34,= -150; 4y"',,3=1450; 



4y' 



534 



4y"',3,=:150; 

 and, therefore, by (216), 



4y'"^3 = 550; 



4y'" — 

 ^* 354 — 



= — 1600 ; 

 400; 



(221) 



(222) 



(223) 



