On the theory of Gordan-Noether on homogeneous forms with zero Hessian (Improved version).

We give a detailed proof for Gordan-Noether's results in ``Ueber die algebraischen Formen, deren Hesse'sche Determinante identisch verschwindet'' published in 1876 in Mathematische Annahlen. C. Lossen has written a paper in a similar direction as the present paper, but did not provide a proof for every result. In our paper, every result is proved. Furthermore, our paper is independent of Lossen's paper and includes a considerable number of new observations. An earlier version of this paper has been printed in Proceedings of the School of Science of Tokai University, Vol.49, Mar. 2014. In this version, a serious error has been corrected and some new results have been added.