By William Rowan Hamilton

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1 1 (1, 2, 3, 4, 5)i; α −1 1 1 (1, 2, 3, 4, 5)k; in which ρα1 and ρα are imaginary cube-roots or fifth-roots of unity, according as α1 is 3 or 1 5; while (1, 2, 3, 4, 5)i and (1, 2, 3, 4, 5)k are some two different values of the function f1 , which may be called f1 and f1 , and correspond to different arrangements of x1 , x2 , x3 , x4 , x5 , being also such that f1 f1 α1 = (1, 2, 3, 4, 5)i 1 = a + b(x1 − x2 ) . . (x4 − x5 ), α α1 = (1, 2, 3, 4, 5)k 1 = a − b(x1 − x2 ) . . (x4 − x5 ). α These last equations show that the cube or fifth power (according as a1 is 3 or 5) of the product of (1, 2, 3, 4, 5)i and (1, 2, 3, 4, 5)k is symmetric, and consequently, by what was lately proved, that this product itself is symmetric; so that we may write f1 .

X5 . With these suppositions, the function f1 must, by the principles of a former article, have exactly α1 values, corresponding to changes of arrangement of the five arbitrary quantities x1 , . . x5 ; the exponent α1 must therefore be a prime divisor of the product 120 (= 1 . 2 . 3 . 4 . 5); that is, it must be 2, or 3, or 5. ; and it would always be possible to find symmetric multipliers c1 , c2 , c3 , c4 , which would not all be equal to 0, and would be such that (2) (3) (4) (2) (3) (4) (2) (3) (4) c1 b2 + c2 b2 + c3 b2 + c4 b2 = 0, c1 b3 + c2 b3 + c3 b3 + c4 b3 = 0, c1 b4 + c2 b4 + c3 b4 + c4 b4 = 0; in this manner then we should obtain an equation of the form (2) (3) (4) (2) (3) (4) c1 a1 + c2 a12 + c3 a13 + c4 a14 = c1 b0 + c2 b0 + c3 b0 + c4 b0 + (c1 b1 + c2 b1 + c3 b1 + c4 b1 )xα , in which it is impossible that the coefficient of xα should vanish, because the five unequal values of a1 could not all satisfy one common equation, of the fourth or of a lower degree; we should therefore have an expression for xα of the form xα = d0 + d1 a1 + d2 a12 + d3 a13 + d4 a14 , the coefficients d0 , .

III. . (β, γ, α, δ); IV. . (β, γ, δ, α); and may be denoted by the four characteristics ∇1 , ∇2 , , ∇3 , a,b a,b,c ∇4 ; or more fully by the following, a,b ∇1 , ∇2 , ∇3 , a,b,c ∇4 ; a,b ∇1 implying, when prefixed to any function (α, β, γ, δ), that we are to interchange the ath a,b and bth of the roots on which it depends; ∇2 , that we are to interchange among themselves, a,b,c not only the ath and bth , but also that cth and dth ; ∇3 , that we are to interchange the ath to the bth , the bth to the cth , and the cth to the ath ; namely, by putting that which had been a,b,c bth in the place of that which had been ath , and so on; and finally ∇4 , that the ath is to be changed to the bth , the bth to the cth , the cth to the dth , and the dth to the ath ; so that we have, in this notation, 1,2 I.