Sir William Rowan Hamilton on Equations of the Fifth Degree. 351 



as the following. The biquadratic equation (6), of which the roots are x", x'", 

 x''', x^, shows that, whatever x may be, 



{x — x"){x — x"'){x — x"'){x — x'')= 1 



X* + x'x' + ocfx^ + x'^x + x'* \ (76) 



+ JO (a;'' _|_ x'x + «'*) + «7 (^ + y ) + r ; j 



and, therefore, that 



(^ _ x") (y - y") (or' _ x'") {x' — x'') = 5x'* + 3px" + 2qx' + r. (77) 



If then we multiply the expression (75) by the square of this last function (77), 

 we ought to obtain a symmetric function of all the five roots of the equation of 

 the fifth degree, namely, the product of the ten squares of their differences, mul- 

 tiplied indeed by a numerical coefficient, namely, — 2~'^3~^, as appears from 

 (70) and (71) : and consequently an expression for this product itself, that is for 



{x" — x'''f{x^'-x''y{af" — x"y {x"' — x''f{x'''-x'')\ J 



must be obtained by multiplying the factor 125^* + &c. which is within the 

 brackets in (75), by the square of 5^* + Zpaf^ + 2qx' + r, and then reducing 



by the condition that x'^ + px"^ + qxf^ + /-^ = s. Accordingly this process 



gives : 



p = 3125s^ _ 2,'7bOpqg' 

 + (108/ — gOOpV + 825j!jY + ^OOOpr" + 2250yV) s" 

 — {I2p*qr — ] 6pY — 56qpV + 630p^V + leOO^r' — lOSy*) s 

 + iGpV^ — Ap^'qV — 128pV* + 144j)^V=' + 256r^ — 27?^ ; 



an expression for the product of the squares of the differences of the five roots of 

 an equation of the fifth degree, which agrees with known results. And we see 

 that with this meaning of p, we may write : 



{k^ - \')^ = _ 2-'=' 3-^ p {5x'* + 2>px'^ + Iqx' + r)-\ (80) 



The expression (61) for h^ becomes, therefore : 



J, _ g-2 3-3 5,B r ( f'"- 3'/^'-' + (^' - ^' + 3») A l .81) 



H4_-2 6 5 P(, 5x^' + ?,px'' + 2qaf+r j' ^^^^ 



/x and V having the meanings defined by (63). 



(79) 



