ON A METHOD PROPOSED BY PROFESSOR BADANO FOR THE SOLUTION

OF ALGEBRAIC EQUATIONS

By

William Rowan Hamilton

(Proceedings of the Royal Irish Academy, 2 (1844), pp. 275–276.)

Edited by David R. Wilkins On a Method proposed by Professor Badano for the Solution of Algebraic Equations.

Sir William R. Hamilton

Communicated 4 August 1842.

[Proceedings of the Royal Irish Academy, vol. II (1844), pp. 275–276.]

The President made a communication respecting a method which had been lately proposed by Professor Badano of Genoa, for the solution of algebraical equations of the fifth and higher degrees.*

Lagrange had shown that the function

t^{5} = (x′ + ωx′′ + ω^{2}x′′′ + ω^{3}x^{IV} + ω^{4}x^{IV})^{5}

receives only twenty-four different values, for all possible changes of arrangement of the five quantities, x′, … x^{V}, if ω be an imaginary root of unity, so that

ω^{4} + ω^{3} + ω^{2} + ω + 1 = 0.

Professor Badano has proposed to express these twenty-four values by certain combinations of quadratic and cubic radicals, suggested by the theory of biquadratic equations, and having the following for their type:

t^{5} = H_{1} + √H_{2} + ^{3} \sqrt{H_{3} + √H_{4}} + ^{3} \sqrt{H_{5} − √H_{6}}
+√\{H_{7} + √H_{8} + ^{3} \sqrt{H_{9} + √H_{10}} + ^{3} \sqrt{H_{11} − √H_{12}}\}
+√\{H_{13} + √H_{14} + θ ^{3} \sqrt{H_{15} + √H_{16}} + θ^{2} ^{3} \sqrt{H_{17} − √H_{18}}\}
+√\{H_{19} + √H_{20} + θ^{2} ^{3} \sqrt{H_{21} + √H_{22}} + θ ^{3} \sqrt{H_{23} − √H_{24}}\};

θ being here an imaginary cube-root of unity. He contends that the twenty-four quantities, H_{1}, … H_{24}, are all symmetric functions of the five quantities x′, … x^{V}; and that they are connected among themselves by the sixteen relations

H_{3} = H_{5}, H_{4} = H_{6}, H_{7} = H_{13} = H_{19}, H_{8} = H_{14} = H_{20}, H_{9} = H_{15} = H_{21},

H_{10} = H_{16} = H_{22}, H_{11} = H_{17} = H_{23}, H_{12} = H_{18} = H_{24}, H_{9} = H_{11}, H_{10} = H_{12}.

* Nuove Ricerche sulla Risoluzione Generale delle Equazioni Algebriche del P. Gerolamo Badano, Carmelitano scalzo, Professore di Matematica nella R. Universita di Genova. Genova, Tipografia Ponthenier, 1840. See also an ‘Appendice’ to the same work.

Sir W. Hamilton examines, in great detail, the composition of the two conjugate quantities H_{4}, H_{6}, which are each of the thirtieth dimension relatively to the five original quantities x′, … x^{V}; though each is symmetric relatively to four of them. He finds also that these two quantities H_{4} and H_{6} are not generally equal to each other, but differ by the sign of an imaginary radical, namely,

(θ − θ^{2})(ω − ω^{2} − ω^{3} + ω^{4}) = \sqrt{−15},

when they are fully developed, in consistency with Professor Badano’s definitions. Analogous results are obtained for the two quantities H_{3}, H_{5}; and these general results are verified by applying them to a particular system of numerical values of the five quantities x′, … x^{V}. It is shown also that the three quantities H_{7}, H_{13}, H_{19}, are neither independent of the arrangement of those five quantities x, nor (generally) equal to each other. And thus, although H_{1} is symmetric, and H_{2} vanishes, Sir W. H. conceives it to be proved that Professor Badano’s expressions, for the twenty-four values of Lagrange’s function t^{5}, give no assistance towards the solution of the general equation of the fifth degree, and therefore that the same method could not be expected to resolve equations still more elevated, even if we were not in possession of an à priori proof that no root of any general equation above the fourth degree can be expressed as a function of its coefficients, by any finite combination of radicals and rational functions.