From page 31 of “A Treatise on the Higher Plane Curves” by George Salmon, Cambridge, 1879
44. We call the deficiency of a curve the number D, by which its number of double points is short of the maximum; this number playing a very important part in the theory of curves. If D = 0, that is, if a curve have its maximum number of double points, the coordinates of any point on the curve can be expressed as rational algebraic functions of a variable parameter. For the ½(n − 1)(n − 2) double points, and n − 3 other assumed points on the curve, making together ½(n + 1)(n − 2) − 1 points, or one less than enough to determine a curve of degree n − 2, we can describe through these points a system of such curves included in the equation U = λV. Now if we eliminate either variable between this equation and that of the given curve, we get to determine the other coordinate for their points of intersection, an equation of the n(n − 2) degree in which λ enters in the nth degree. But of this equation all the roots but one are known; for the intersections of the curves consist of the double points counted twice, of the n − 3 assumed points, and only of one other point, since

(n − 1)(n − 2) + (n − 3) + 1 = n(n − 2).

Dividing out, then, the known factors of the equation, the only unknown root remains determined as an algebraic function of the nth degree in λ.

It is true, conversely, that if the coordinates can be expressed as rational functions of a parameter, the curve has the maximum number of double points, Curves of this sort are called unicursal curves. When we are given x, y, z respectively proportional to aλn + &c., a'λn + &c., a''λn + &c., the actual elimination of λ is easily performed dialytically. Writing down the three equations

θx = aλn + &c., θy = a'λn + &c., θz = a''λn + &c.,

and multiplying each successively by λ, λ2, … λn−1, we shall have 3n equations, exactly enough to eliminate linearly all the quantities θ, θλ, &c., λ, λ2, &c. The equation of the curve, then, appears in the form of a determinant of the order 3n, but only n rows will contain the variables; the curve therefore will be of the nth order, and its equation will involve the coefficients a, b, &c., in the 2nth degree. All this will be more clearly understood if we actually write down the result for the case n = 2. We have, then, the three equations

θx = aλ2 + bλ + c, θy = a'λ2 + b'λ + c', θz = a''λ2 + b''λ + c''.

Multiplying each by λ, and then eliminating linearly from the six equations the quantities θ, θλ, λ3, λ2, λ, the result appears as the determinant
x0abc0
y0a'b'c'0
z0a''b''c''0
0x0abc
0y0a'b'c'
0z0a''b''c''
= 0
This is the same as the final equation, Higher Algebra, Art. 193.