terms, and was led to an important determinant involving the second differential coefficient of a form of the third degree, called the "Hessian."
The "Hessian" plays a leading part in the theory of invariants, a subject first studied by Cayley. Hesse showed that his determinant gives for every curve another curve, such that the double points of the first are points on the second, or "Hessian." Similarly for surfaces (Crelle, 1844). Many of
the most important theorems on curves of the third order are due to Hesse. He determined the curve of the 14th order, which passes through the 56 points of contact of the 28 bi-tangents of a curve of the fourth order. His great memoir on this subject (Crelle, 1855) was published at the same time as was a paper by Steiner treating of the same subject.
Hesse's income at Königsberg had not kept pace with his growing reputation. Hardly was he able to support himself and family. In 1855 he accepted a more lucrative position at Halle, and in 1856 one at Heidelberg. Here he remained until 1868, when he accepted a position at a technic school in Munich.67 At Heidelberg he revised and enlarged upon his
previous researches, and published in 1861 his Vorlesungen über die Analytische Geometrie des Raumes, insbesondere über Flächen 2. Ordnung. More elementary works soon followed. While in Heidelberg he elaborated a principle, his "Uebertragungsprincip." According to this, there corresponds to every point in a plane a pair of points in a line, and the projective geometry of the plane can be carried back to the geometry of points in a line.
The researches of Plücker and Hesse were continued in England
by Cayley, Salmon, and Sylvester. It may be premised here
that among the early writers on analytical geometry in England was James Booth (1806–1878), whose chief results are embodied
in his Treatise on Some New Geometrical Methods; and James MacCullagh (1809–1846), who was professor of natural philosophy
at Dublin, and made some valuable discoveries on the theory of quadrics. The influence of these men on the progress of geometry was insignificant, for the interchange of scientific results between different nations was not so complete at that time as might have been desired. In further illustration of this, we mention that Chasles in France elaborated
subjects which had previously been disposed of by Steiner in
Germany, and Steiner published researches which had been given by Cayley, Sylvester, and Salmon nearly five years earlier. Cayley and Salmon in 1849 determined the straight lines in a cubic surface, and studied its principal properties, while Sylvester in 1851 discovered the pentahedron of such a surface. Cayley extended Plücker's equations to curves of higher singularities. Cayley's own investigations, and those of M. Nöther of Erlangen, G. H. Halphen (1844–1889) of the
Polytechnic School in Paris, De La Gournerie of Paris, A. Brill