56 Lectures on Scientific Subjects
p. Hence in this case the density function at any point is
simply given by the sum of the reciprocals of the lengths of
tangents which can be drawn from the given point to the
circles and circular arcs.
Our second case of a one-parameter family will be the
degenerate case in which the curve C enveloped by the
straight lines reduces to a single point. For definiteness we
will further assume that the frequency of the points of inter-
section of the radial lines through this point O is given by a
continuous function/(ç?) on the unit circle about the point O;
here φ designates the angle which the lines make with a
fixed line. It is then apparent that we have F(r, S) =f(,φ)∕r
as the corresponding density function, and that this density
function is continuous everywhere except at the origin
r = 0, where it is obviously discontinuous.
The third and last (one-parameter) case is the completely
degenerate case in which there is no envelope and we have a
family of parallel straight lines so that F(r, S) ≈f(r cosθ) if all
the lines make an angle of τr∕2 with the initial direction.
If we had employed corresponding rectangular coordinates
x, yt we would have
F(r, S') = F*{x, y) =f(F),
or more generally, for an arbitrary direction of the family of
parallel straight lines, we would have
F*(,τ, y) =f(ax+by).
Here f(ax+by) is taken as an arbitrary continuous function
of its argument ax+by.
Now it has long been known that an arbitrary continuous
function of x and y can be approximated to as a sum of
continuous functions of such a linear variable ax+by. Con-
sequently it is obvious that any positive continuous function