By William Rowan Hamilton

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On the other hand, no expression of the form b0 + b1 a1 can represent the three-valued function x; we must therefore suppose that if the sought expression b(m) exist at all, it is, at lowest, of the second order, and involves at least one radical a1 , such that a1 α1 = ( f1 = ) b 0 + b 1 a 1 , and a1 = f1 (x1 , x2 , x3 ); the rational function f1 admitting of 2α1 values, and consequently the exponent α1 being = 3, (since it cannot be = 2, because no function of three variables has exactly four values,) so that we must suppose the radical a1 to be a cube-root, of the form a1 = 3 b0 + b1 a1 , b0 and b1 being rational with respect to a1 , a2 , a3 .

B2 . (e21 − e32 ) = −214 . 33 . b2 . (e21 − e32 ), b being some symmetric function of x1 , x2 , x3 , x4 , and e1 , e2 having the same meanings here as in the second article; because no rational and unsymmetric function of four arbitrary quantities x1 , x2 , x3 , x4 , has a prime power symmetric, except either this function a1 , or else some other function a2 which may be deduced from it by a multiplication such as the c following, a2 = a1 . But a two-valued expression of the form f1 = b0 + b1 a1 cannot represent b a four-valued function, such as x; we must therefore suppose that the sought expression b(m) contains a radical a1 of the second order, and this must be a cube-root, of the form a1 = (p0 + p1 a1 )(u1 + ρ23 u2 + ρ3 u3 ) = √ 3 (b0 + b1 a1 ); in which ρ3 is, as before, an imaginary cube-root of unity; p0 , p1 , b0 , b1 are symmetric relatively to x1 , x2 , x3 , x4 , or rational relatively to a1 , a2 , a3 , a4 ; u1 = x1 x2 + x3 x4 , u2 = x1 x3 + x2 x4 , u3 = x1 x4 + x2 x3 ; and b0 + b1 a1 = 1728(p0 + p1 a1 )3 e1 + 1 a (ρ23 − ρ3 ) 1 1152 b , the rational function e1 , and the radical a1 retaining their recent meanings: because no rational function f1 of four independent variables x1 , x2 , x3 , x4 , which cannot be reduced to the form thus assigned for a1 , can have itself 2α1 values, α1 being a prime number greater α than 1, if the number of values of the prime power f1 1 be only 2.

X3 − x5 ), which is an equation of the form (b) , and reduces the function (b) to the form (a), and ultimately to a symmetric function a, because x5 and x4 may be interchanged. The supposition (b) conducts to a two-valued function, which changes value when any two of the five roots are interchanged, so that the sum (1, 2, 3, 4, 5) + (1, 2, 3, 5, 4), and the quotient (1, 2, 3, 4, 5) − (1, 2, 3, 5, 4) , (x1 − x2 )(x1 − x3 ) . . x4 − x5 , 52 in which a and b are symmetric. The remaining suppositions, (c) , (c) , (d) , (d) , are easily seen to conduct only to symmetric functions; for instance, (c) gives φ(x5 , x1 x2 + x3 x4 ) = φ(x4 , x3 x5 + x2 x1 ) = φ(x1 , x3 x5 + x2 x4 ) = φ(x1 , x2 x4 + x3 x5 ) = φ(x5 , x2 x4 + x3 x1 ) = φ(x5 , x1 x3 + x2 x4 ), so that the condition (c) is satisfied, and at the same time x5 is interchangeable with x4 .

### On the Argument of Abel by William Rowan Hamilton

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