and were partly synthetic. He solved the problem to construct any tenth point of such a surface when nine points are given. The analogous problem for a conic had been solved by Pascal by means of the hexagram. A difficult problem confronting mathematicians of this time was that of elimination. Plücker had seen that the main advantage of his special method in analytic geometry lay in the avoidance of algebraic elimination. Hesse, however, showed how by determinants to make algebraic elimination easy. In his earlier results he was anticipated by Sylvester, who published his dialytic method of elimination in 1840. These advances in algebra Hesse applied to the analytic study of curves of the third order. By linear substitutions, he reduced a form of the third degree in three variables to one of only four 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
