To develop the theory, consider the curve corresponding to any particular value of the parameter; this has with the consecutive curve (or curve belonging to the consecutive value of the parameter) a certain number of intersections and of common tangents, which may be considered as the tangents at the intersections; and the so-called envelope is the curve which is at the same time generated by the points of intersection and enveloped by the common tangents; we have thus a dual generation.
It is obvious that the co-ordinates of any point on an ellipse may be expressed in terms of a single parameter, the abscissa being a cos q4, and the ordinate b sin 43, since on eliminating 4 between x = a cos and y = b sin 4) we obtain the equation to the ellipse.
Inflections, in, n, 0, being connected by the Pluckerian equations, - the number of nodes or cusps may be greater for particular values of the parameter, but this is a speciality which may be here disregarded.
The parameter which determines the variable curve may be given as a point upon a given curve, or say as a parametric point; that is, to the different positions of the parametric point on the given curve correspond the different variable curves, and the nature of the envelope will thus depend on that of the given curve; we have thus the envelope as a derivative curve of the given curve.
The conics are distinguished by the ratio between the latus rectum (which was originally called the latus erectum, and now often referred to as the parameter) and the segment of the ordinate intercepted between the diameter and the line joining the second vertex with the extremity of the latus rectum.
The notion is very probably older, but it is at any rate to be found in Lagrange's Theorie des fonctions analytiques (1798); it is there remarked that the equation obtained by the elimination of the parameter a from an equationf (x,y,a) = o and the derived equation in respect to a is a curve, the envelope of the series of curves represented by the equation f (x,y,a) = o in question.
Considering the variable curve corresponding to a given value of the parameter, or say simply the variable curve, the consecutive curve has then also 6 and nodes and cusps, consecutive to those of the variable curve; and it is easy to see that among the intersections of the two curves we have the nodes each counting twice, and the cusps each counting three times; the number of the remaining intersections is = m 2 - 263 K.
=o, we may start with an equation u=o, where u is a function of the order m containing a parameter 0, and for a particular value say 0=o, of the parameter reducing itself to Piaip2a2....
If, on the other hand, the sulphur system be replaced by a corresponding selenium system, an element of higher atomic weight, it would be expected that a slight increase would be observed in the vertical parameter, and a greater increase recorded equally in the horizontal parameters.
In fact in a unicursal curve the co-ordinates of a point are given as proportional to rational and integral functions of a parameter, so that any point of the curve is determined uniquely by means of this parameter; that is, to each point of the curve corresponds one value of the parameter, and to each value of the parameter one point on the curve; and the (a, t3) correspondence between the two points is given by an equation of the form MO, I) u (4), 01 3 =0 between their parameters 0 and 4); at a united point 4)=0, and the value of 0 is given by an equation of the order a+ 0.