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



posed equation (2), and so to obtain Hj as a rational function of the coefficients 

 of that equation, namely, 



125 



H. = -Y2-(i'?-25«). (51) 



Indeed, it was evident a priori that h, must be found to be equal to some ra- 

 tional function of those four coefficients, p, q, r, s, or some symmetric function 

 of the five roots of the equation (2) ; because it is, by its definition, the sixth 

 part of the sum of the six functions v, and, therefore, the twenty-fourth part of 

 the sum of the twenty-four different values of the function t ; or finally the mean 

 of all the different values which the function f' can receive, by all possible changes 

 of arrangement of the five roots y, . . ^^, or jr,, . . x^, among themselves. The 

 evanescence of h^ shows farther, that, in the arrangement assigned above, the sum 

 of the three first of the six functions v, or the sum of the twelve first of the 

 twenty-four functions t, is equal to the sum of the other three, or of the other 

 twelve of these functions. But we shall find that it would be erroneous to con- 

 clude, from the analogy of these results, even when combined with the corres- 

 ponding results for equations of Inferior degrees, that the other four quantities 

 H, which enter into the formulas (a) and (b), can likewise be expressed as ra- 

 tional functions of the coefficients of the equation of the fifth degree. 



19. The auxiliary quantities b^ c% d% being seen, by (46), to be the three 

 roots »„ z^, z^ of the cubic equation 



z'+5z^— 45« — 25 = 0, (52) 



which decomposes itself into one of the first and another of the second degree, 



namely, 



z — 5 = 0, z^-^10z-\-5 = 0; (53) 



we see that one of the three quantities b, c, d, must be real, and =z ± V5, 

 while the other two must be imaginary. And on referring to the definitions 

 (30), and remembering that w is an imaginary fifth root of unity, so that w* and 

 w' are the reciprocals of w and w\ we easily perceive that the real one of the 

 three is d, and that the following expressions hold good : 



B^zz— 5— 2d; c'= — 5-f-2D; d* = 5; (54) 



with which we may combine, whenever it may be necessary or useful, the rela- 

 tion 



