54 Lectures on Scientific Subjects
in the same plane by the coordinates ʃ and φ as indicated in
the figure. These are the usual line coordinates useful for
our purposes.
2. ONE-PARAMETER FAMILIES OF STRAIGHT LINES
Let us imagine now that a family of straight lines be
drawn which depends on a single parameter. We consider
first the following case: the straight lines envelop a regular
convex arc C with continuous curvature l∕p, along which
the points of tangency of the lines are distributed with an
assigned frequency/(ʃ), where ʃ designates arc length along
the curve C (see figure 3).
c
Fig. 3
This means that to get an approximation to the limiting
form of drawing we may distribute the tangent lines along
the curve C at points /(ʃ) ds apart, (ds, a small fixed in-
crement of arc). It is obvious that the function /(ʃ) is a
kind of distribution function, specifying the distribution of
the given one-parameter family of straight lines.