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



H, = _2-«3'5'«7'(497 — 132^): (197) 



results which differ from those obtained with the former arrangement of the five 

 roots of the proposed equation (147), but of which the agreement with the ge- 

 neral formulae of the present paper may be evinced by processes similar to those 

 of the last article. 



39. As a last example, if the arrangement of the same five roots be 



X, = 0, a,\ =1, Xj = 2, x^ = — 1, JTs = _ 2, (198) 



we then find easily that all the six quantities v vanish, and, therefore, that we 

 have, with this arrangement, 



\/h4 = 0, H4 = 0. (199) 



All these results respecting the numerical values of H4, for different arrange- 

 ments of the roots of the proposed equation (147), are Included in the common 

 expression : 



H,__2 3 5 (^ 5x''-l5x'-'+l J' (^"^> 



which results from the formula (85), combined with (79) and (86) (87) (88) : 

 and thus we have a new confirmation of the correctness of the foregoing calcula- 

 tions. 



40. It is then proved, in several different ways, that the quantity h^, in the 

 formulae which have been marked in this paper (a), (b), (c), (d), and which have 

 been proposed by Professor Badano for the solution of the general equation of 

 the fifth degree, is not a symmetric function of the five roots of that equation. 

 And since it has been shown that the expression of this quantity h^, contains in 

 general the imaginary radical ^ or \/ — 15, which changes sign in passing to the 

 expression of the analogous quantity Hg, we see that these two quantities, h^ and 

 ^g, are not generally equal to each other, as Professor Badano, in a supplement 

 to his essay, appears to think that they must be. They are, on the contrary, 

 found to be in general the two unequal roots of a quadratic equation, namely, 



h/ + QH, + K* = 0, (201) 



in which 



Q = - (h, + hJ = 2-" 3-^ 5'*w^ (5l'^ - 3l"*), (202) 



and 



B = Vuy Va, = — 2-'" 3-' 5'' ^"^ (5l'* + 3l"^), (203) 



