By Wiliam Rowan Hamilton

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To show that the constancy, thus asserted, does not exist, has been the chief object proposed in the present paper; to which the writer has had opportunities of making some additions, since it was first communicated to the Academy.

The twelve other expressions which have been referred to, as being analogous to (a) and (b), are of the forms: 2 v345 = h7 + 2 v354 2 v534 2 v435 = h7 − √ √ h8 + 3 h9 + √ √ h10 + 3 h11 − √ √ h12 ; h8 + h9 − h10 + h11 + h12 ; √ √ √ = h13 + h14 + 3 h15 + h16 + 3 h17 − h18 ; √ √ √ = h13 − h14 + 3 h15 − h16 + 3 h17 + h18 ; 3 3 (a ) (b ) (a ) (b ) and they give, as the simplest of the expressions deduced from them, the two following, which are analogous to that marked (220): 2 2 2 2 2 2 h7 = 16 (v345 + v453 + v534 + v354 + v543 + v435 ); h13 = 2 1 6 (v345 + 2 v453 + 2 v534 + 2 v354 + 2 v543 + 2 v435 ).

X5 . The expression (101) for h4 itself is therefore reproduced under the form: h4 = −2−2 3−3 518 2 m21 m22 m23 ; (146) and the conclusions of former articles are thus confirmed anew, by a method which is entirely different, in its conception and in its processes of calculation, from those which were employed before. 33. It may not, however, be useless to calculate, for some particular equation of the fifth degree, the numerical values of some of the most important quantities above considered, and so to illustrate and exemplify some of the chief formulæ already established.

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On Equations of the Fifth Degree by Wiliam Rowan Hamilton


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