It is an old and easily proved theorem that, for a curve of the order m, the number δ + κ of nodes and cusps is at most = ½(m − 1)(m − 2); for a given curve the deficiency of the actual number of nodes and cusps below this maximum number, viz. ½(m − 1)(m − 2) − δ − κ, is the “Geschlecht” or “deficiency,” of the curve, say this is = D. When D = 0, the curve is said to be unicursal, when = 1, bicursal, and so on.
The general theorem is that two curves corresponding rationally to each other have the same deficiency. [In particular a curve and its reciprocal have this rational or (1, 1) correspondence, and it has been already seen that a curve and its reciprocal have the same deficiency.]
A curve of a given order can in general be rationally transformed into a curve of a lower order; thus a curve of any order for which D = 0, that is, a unicursal curve, can be transformed into a line; a curve of any order having the deficiency 1 or 2 can be rationally transformed into a curve of the order D + 2, deficiency D; and a curve of any order deficiency = or > 3 can be rationally transformed into a curve of the order D + 3, deficiency D.
Taking x′, y′, z′ as co-ordinates of a point of the transformed curve, and in its equation writing x′ : y′ : z′ = 1 : θ : φ we have φ a certain irrational function of θ, and the theorem is that the co-ordinates x, y, z of any point of the given curve can be expressed as proportional to rational and integral functions of θ, φ, that is, of θ and a certain irrational function of θ.
In particular if D = 0, that is, if the given curve be unicursal, the transformed curve is a line, φ is a mere linear function of θ, and the theorem is that the co-ordinates x, y, z of a point of the unicursal curve can be expressed as proportional to rational and integral functions of θ; it is easy to see that for a given curve of the order m, these functions of θ must be of the same order m.
If D = 1, then the transformed curve is a cubic; it can be shown that in a cubic, the axes of co-ordinates being properly chosen, φ can be expressed as the square root of a quartic function of θ; and the theorem is that the co-ordinates x, y, z of a point of the bicursal curve can be expressed as proportional to rational and integral functions of θ, and of the square root of a quartic function of θ.
And so if D = 2, then the transformed curve is a nodal quartic; φ can be expressed as the square root of a sextic function of θ and the theorem is, that the co-ordinates x, y, z of a point of the tricursal curve can be expressed as proportional to rational and integral functions of θ, and of the square root of a sextic function of θ. But D = 3, we have no longer the like law, viz. φ is not expressible as the square root of an octic function of θ.
Observe that the radical, square root of a quartic function, is connected with the theory of elliptic functions, and the radical, square root of a sextic function, with that of the first kind of Abelian functions, but that the next kind of Abelian functions does not depend on the radical, square root of an octic function.
It is a form of the theorem for the case D = 1, that the co-ordinates x, y, z of a point of the bicursal curve, or in particular the co-ordinates of a point of the cubic, can be expressed as proportional to rational and integral functions of the elliptic functions snu, cnu, dnu; in fact, taking the radical to be √1 − θ²·1 − k²θ², and writing θ = snu, the radical becomes = cnu, dnu; and we have expressions of the form in question.
It will be observed that the equations x′ : y′ : z′ = X : Y : Z before mentioned do not of themselves lead to the other system of equations x : y : z = X′ : Y′ : Z′, and thus that the theory does not in anywise establish a (1, 1) correspondence between the points (x, y, z) and (x′, y′, z′) of two planes or of the same plane; this is the correspondence of Cremona’s theory.