Sir William Rowan Hamilton on Equations of the Fifth Degree. 367 



M.= -^-(3 + 4^); M, = A.(3_^); M3=f; (191) 



and which conduct still to the same result. 



38. An equation of the fifth degree, which, like that here assumed as an ex- 

 ample, has all its roots unequal, may have those roots arranged in 120 different 

 ways ; and any one of these arrangements may be taken as the basis of a verifica- 

 tion such as that contained in the last five articles. But we have seen that no 

 such change of arrangement will affect the value of either H, or Hj ; and with 

 respect to H4, which has been more particularly under our consideration in this 

 paper, it is not difficult to perceive that an interchange of any two of the four 

 last roots (.r^, x^, x^, x^, or x", x'", x^^, x^), of the proposed equation of the 

 fifth degree, will merely change the sign of the square-root, a/h^, in the fore- 

 going formulte, without making any change in the value of H4 itself, which has 

 been shown to depend on the first root (^, or x') alone. It will, however, be 

 instructive to exemplify this last-mentioned dependence, by applying the fore- 

 going general processes to the case of the equation of the fifth degree (147), the 

 two first roots being made to change places with each other, in such a manner 

 that the order shall now be chosen as follows : 



X, = 1, s, - 2, x^ = 0, x^ =z-l, x, = - 2, (192) 



or (since the sum of all five vanishes), 



x' = l, x" = 2, x"'=zO, x"'=-l, x''= — 2. (193) 



We find, for this new case, by calculations of the same sort as in recent articles 

 of this paper, the following new system of equations for the values of the six 

 functions v : 



Ti7V34, = 12 + 4D; ^|.^v,,3 = -9-4d; ^|^v,3, = _3; j 



tItV435 = 12 — 4d ; TfTV543 = — 9 + 4d ; t|-3-^'354 = — 3 ; J 



in which, d has again the meaning assigned by (30) : and, consequently, 



S (V345 + eV,,3 + ew,,0 = 3 (50^ _ 26) - 4f ; 1 



rh (V3M + eV,„ + ev«5) = 3 (56 -26') -4^; J 



2' 3' 5-'' Vh, = {3 (50-^ _ 26) — 4f }^ — {3 {56 — 26') _ 4^Y ; ] 



VH,= 2-'5m6-6'){55-60', 1 



and 



(194) 



(195) 

 (196) 



