Zeuthen, Geschichte der Mathematik im 16.
Zeuthen, Die Lehre von den Kegelschnitten im Altertum (Copenhagen, 1886 and 1902).
Zeuthen that at most eight of these are real.
It was assumed by Plucker that the number of real double tangents might be 28, 16, 8, 4 or o, but Zeuthen has found that the last case does not exist.
But although we thus arrive by projection at the notion of a circuit, it is not necessary to go out of the plane, and we may (with Zeuthen, using the shorter term circuit for his complete branch) define a circuit as any portion (of a curve) capable of description by the continuous motion of a point, it being understood that a passage through infinity is permitted.
It was thus that Zeuthen (in the paper Nyt Bydrag, " Contribution to the Theory of Systems of Conics which satisfy four Conditions " (Copenhagen, 1865), translated with an addition in the Nouvelles Annales) solved the question of finding the characteristics of the systems of conics which satisfy four conditions of contact with a given curve or curves; and this led to the solution of the further problem of finding the number of the conics which satisfy five conditions of contact with a given curve or curves (Cayley, Comptes Rendus, t.
He determines in every case the characteristics (µ, v) of the corresponding systems of cubics (4p), (3 p, il), &c. The same problems, or most of them, and also the elementary problems in regard to quartics are solved by Zeuthen, who in the elaborate memoir " Almindelige Egenskaber, &c.," Danish Academy, t.
The methods of Maillard and Zeuthen are substantially identical; in each case the question considered is that of finding the characteristics Cu, v) of a system of curves by consideration of the special or degenerate forms of the curves included in the system.
Zeuthen in the case of curves of any given order establishes between the characteristics pc, v, and 18 other quantities, in all 20 quantities, a set of 24 equations (equivalent to 2 3 independent equations), involving(besides the 20 quantities) other quantities relating to the various forms of the degenerate curves, which supplementary terms he determines, partially for curves of any order, but completely only for quartic curves.
Zeuthen, "Abzahlende Methoden," Bd.
Zeuthen, Die Lehre von dem Kegelschnitten in Alterthum (1886).