Sir William Rowan Hamilton on Equations of the Fifth Degree. 337 



It may be noticed. also that the present method gives for the product of these 

 three square roots, the expression : 



y/z'. ^Z".^Z"' = ^ (X' + X" - X'" — X'") {X' - X" + X'" - x'") 



(^a;'_a:"-x"'^x"') 



= (_ 2a 4- /3 + 7) (— 2a + ej3 + O'y) ( —2a + 6'p -\- Oy) 



= — 8a^ + 6ar] 4" 2e = — q ; 



a result which may be verified by observing that, by the expressions given above 

 for a, t]', e', in terms of a, 7], e, we have the relation 



z'z"z"' = a'' — 3a r,' + 2e' = (— 8a^ + Barj + 2e)^ 



12. In this manner, then, it might have been discovered that the four roots 

 :i'„ X2, Xp x^, of the general biquadratic equation 



X* — Ax^ -\- Bx'' — ex -\- T> =^0, 



are the four values of an expression of the form a -|- 13 + 7 -{- 8, in which, a, /3- -|- 

 7- -j- 8', /378, and )3^7^ -j- y'i- + c-^'; are rational functions of the coefficients 

 A, B, c, D, and may be determined as such by comparison with the identical 

 equation 



(a + ^ + 7+S_a)^-2(p^ + 7^ + 8^)(a + ^ + 7+5-a)^ 



+ (/3' + r + ^7 = 8i37K« + ^ + 7 + S - a) + 4 (^y + 7^8^ + 8-'/30, 



of which each member is an expression for the square of 2 (^y + 78 + ^P)- It 

 might have been perceived also that any three quantities, such as here /S'-, y\ 8', 

 which are the three roots of a given cubic equation, may be considered as the 

 three values of an expression of the form a -}- ^ -\- y', in which, a', ^'y, and 

 ^^ -f 7'^ are rational functions of the coefficients of that given equation, and may 

 have their forms determined by comparison witli the identity, 



(*' + ^ + 7 - «■')' - 3py' {a! + ^' 4- 7' _ a') - ^" - 7'' = 0. 



And finally that any two quantities which, as here /3'^ and 7', arc the two roots 

 of a given quadratic equation, are also the two values of an expression of the form 

 a" -f /3 ', in which a" and ^'^ may be determined by comparing the given equa- 

 tion with the following identical form, 



(a" + p" _ cc"f - ^"-^ = 0. 

 VOL. XIX. 2 X 



