The name is absent

58 Lectures on Scientific Subjects

lines of breadth proportional to

f(μ∆s, r∆^)∆∫∆∣ρ (μ, V = 1, 2, ∙ ∙ ∙n),
are drawn (i.e., /(μ∆j, v∆φ) lines of the same breadth).
The kind of distribution of lines just specified is essentially
independent of both the origin of coordinates and the
direction of the initial line. It is clear that in a small area dA
the limiting density as n increases without limit will be
given by

f_o f{s, φ)dφ.

Now we have the relationship (see fig. 2) s=r sin(<p- θ~).
In consequence we have the following general fundamental
equation connecting the density function F(r, θ) and the
distribution function /(ʃ, φ) :

(1)            T(r, O)          sin(ρ - θ), φ)dφ.

Thus, given the continuous distribution function /(ʃ, φ),
say for И ≤S (S < + ∞) it is immediately possible to deter-
mine the corresponding continuous density function F(r, θ^)
for ∣r∣ ≤S by the above formula.

Our primary concern is of course the inverse problem:
Given a density function F(r, θ) for ∣r∣ ≤R < + ∞, to ascer-
tain whether or not there exists a corresponding distribution
function/(ʃ, φ) for ∣j∣ ≤R, and further to determine all such
functions (when they exist). From this point of view the
fundamental equation (1) appears as a special type of
“linear integral equation of the first kind” for the unknown
function /(ʃ, φ). Since we are here concerned with deter-
mining whether or not a given wash drawing can be re-
produced by means of smooth distributions in distance and
angle of very many fine indefinitely extended straight lines,
the solution/(ʃ, ∣p), as well as F(r, θ), is in general required

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