The name is absent

60 Lectures on Scientific Subjects

tribution function /(ʃ, φ), actually involving φ, does yield a
symmetrical density function F(F). It is then clear that
/(ʃ, <p+c) for any c would yield the same density function,
and thus that the average

[ʃ(ʃ, φ)+f(s, <p + Δ<p)H-----h/(ʃ, ^> + (n-l)Δ⅛>)]∕n

with n∆<⅛5 = 2τr, would also. Proceeding to the limit, we see
that the symmetrical distribution function

/(ʃ) =f₀^3rf(^s> ^)⅜∕2τr

would also yield the assigned density F(r).

Hence if a поп-symmetrical /(ʃ, φ) did exist, so would a
symmetrical /(ʃ).

As follows from a later result, there cannot exist such a
поп-symmetrical f(s, φ). In fact if there did exist such an
/(ʃ, φ), the difference g(ʃ, φ) =f(s, φ) -f(F) would satisfy
the homogeneous linear integral equation,

0= f_o^2rg(^r sin(<ρ-θ), φ)dφ.

If we write here φ = θ+≠, this equation takes the equivalent
form

O = J_o g(r sin ψ, θ+ψ^)dψ.

But it will be proved that the basic equation (ɪ) admits of a
unique solution / at most; hence in the case F=o, the only
possible one is trivial one ∕=o, and this is the equation
under consideration with∕=g. We observe also that if other
solutions existed it would be possible to make a rectilinear
drawing (non-uniform) and erase it all by rectilinear erasure
along other lines. At present, however, we shall only prove
that at most one symmetrical solution can exist. Otherwise
we obtain the following still simpler equation:

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