Sir William Rowan Hamilton on Equations of the Fifth Degree. 333 

 we know first, by the same general principles, that the number of these values is 



two, because - =: 2, when m = 3, ra := 3. And because these values will not 

 n 



be altered by adding any common term to the three roots a/, x", x'", it is per- 

 mitted to treat the sum of these three roots as vanishing, or to assume that 



x' + x" + x'" = ; 



that is, to reduce the cubic equation to the form 



x'^ -|- px' -{■ q-=zO. 



In other words, the function 



e={x, + ex,-^6'x,y, 



in which x^, x^ x^ are the three roots of the equation with coefficients a, b, c, 

 will depend on those coefficients, only by depending on p and q, if these two 

 quantities be chosen such that we shall have identically » 



ar* — A^'^ -\-'&x — c — {x — \ kf ■\- f {x — :j a) -1" §-. 



8. This being perceived, and x" and x'" being seen to be the two roots of 

 the quadratic equation 



y^+yy + y^ + p = o, 



which is obtained by dividing the cubic 



. x"^-\-px" — x'^ -px' = Q, 



by the linear factor xf' — x' \ we may, by the theory of quadratics, assume the 

 expressions 



x" = a-{-p, x"' = a—p, 



provided that we make 



a=-l-x', ^ = -^x"-p, 



that is, provided that we establish the identity 



(x" - af — ^ = x"' + X' X" + x'^ + p. 



And, substituting for x', x", x'", their values as functions of a and /3, and reduc- 

 ing by the equation 0^ + -f- 1 = 0, we find 



