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



we see that the six functions v may be expressed by the help of square-roots 

 and cube-roots, in terms of these six quantities h, by means of the following for- 

 mulae : 



V345= Hi+ V^H^ + \/h3 + \/h, -f \/h. 



'v/h.; 



V453 = H, + \/Hj + 0A/H3-f \/h,+ 0Vh, 



Vh^; 



Vm4 = Hi + V'h^ + 0VH3 4-\/H4-f eVn^ 



Vh^; 



(a) 



and 



'354 



= H, — \/H2-f V'Hj — \/h, + V'Hj + V'He; 



, = H, - Vh, + 6 Vn^ - Vh, + e"- Vh, + Vhb ; 



H, — Vh^-^-OWh^ — Vh^+ Va^-\- Vhq-. 



(b) 



which have accordingly, with some slight differences of notation, been assigned 

 by Professor Badano, as among the results of his method of treating equations 

 of the fifth degree. We see, too, that the six quantities h,, . . . h„, (of which in- 

 deed the second, namely, u^, vanishes), are rational functions of a, e, rj, t; and 

 therefore, by article 13., of .r', p, q, r. But it is necessary to examine whether 

 it be true, as Professor Badano appears to think (guided in part, as he himself 

 states, by the analogy of equations of lower degrees), that these quantities h are 

 all rational functions of the coefficients jo, y, r, s, of the equation (2) of the fifth 

 degree ; or, in other words, to examine whether it be possible to eliminate from 

 the expressions of those six quantities h, the unknown root .r' of that equation, by 

 its means, in the same way as it was found possible, in articles 11. and 9- of the 

 present paper, to eliminate from the correspondent expressions, the roots of the 

 biquadratic and cubic equations which it was there proposed to resolve. For, if 

 it shall be found that any one of the six quantities h,, . . . h^, which enter into the 

 foriTiulae (a) and (b), depends essentially, and not merely in appearance, on the 

 unknown root jc'; so as to change its value when that root is changed to another, 

 such as x", which satisfies the same equation (2) : it will then be seen that these 

 formulze, although true, give no assistance towards the general solution of the 

 equation of the fifth degree. 



18. The auxiliary quantities w, b, c, d, being such that, by their definitions 

 (20) and (30), 



