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Published online by Cambridge University Press: 05 July 2011
The equations referred to are: the first of them, that given by Jacobi in the paper “Observatiunculae ad theoriam aequationum pertinentes,” Crelle, t. xiii. (1835), pp. 340–352, [Ges. Werke, t. iii., pp. 269–284], under the heading “Observatio de aequatione sexti gradus ad quam aequationes sexti gradus revocari possunt,” and the second, that of Kronecker in the note “Sur la résolution de l'équation du cinquième degreé,” Comptes Rendus, t. xlvi. (1858), pp. 1150–1152. Jacobi's equation is closely connected with that obtained by Malfatti in 1771, see Brioschi's paper “Sulla resolvente di Malfatti per l'equazione del quinto grado,” Mem. R. Ist. Lomb., t. ix. (1863); but the characteristic property first presents itself in Jacobi's form, and I think the equation is properly described as Jacobi's resolvent equation. The other equation has been always known as Kronecker's resolvent equation; it belongs to the class of equations for the multiplier of an elliptic function considered by Jacobi in the paper “Suite des notices sur les fonctions elliptiques,” Crelle, t. iii. (1828), pp. 303–310, see p. 308, [Ges. Werke, t. i., pp. 255–263, see p. 261]: say Kronecker's equation belongs to the class of Jacobi's Multiplier Equations. We have in regard to it the paper by Brioschi, “Sul metodo di Kronecker per la resoluzione delle equazioni di quinto grado,” Atti Ist. Lomb., t. i. (1858), pp. 275–282, and see also the “Appendice terza” to his translation of my Elliptic Functions (Milan, 1880): it seems to me however that the theory of Kronecker's equation has not hitherto been exhibited in the clearest form.
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