Rectilinear Drawing 65
It is easily established that such a continuous harmonic
distribution function of the mth order f(s, φ) yields always a
corresponding continuous harmonic distribution function of
the mth order, F(r, 0).
In fact, from (1) we obtain directly
F(r, θ^) =J (fm(r sin (⅞s- θ)) cos mφ
+gm(r sin (tp-θf) sin mφ)dφ
= sin ×) cos 7w(x + θ)
+gm(r sin χ) sin m(χ+θ')')dχ
= [J'oa,r(‰0' sin χ) cos mχ
+gm(r sin χ) sin mχ) dχ^ cos mθ
+ [.{"(-AO sin χ)sin mX
÷gm(r sin χ) cos mχ) dχ^ sin mθ.
Conversely we can at least conclude that if for a given har-
monic density function Fm{r, 0) of order m there is a corre-
sponding continuous distribution function f(s, φ), this will
necessarily be unique and also harmonic of order m.
To establish this uniqueness, suppose that /(ʃ, φ) is a
distribution function yielding the harmonic Fm(r, θ) as cor-
responding density function. The expansion off(s, φ) in a
Fourier series breaks f(s, φ) up (formally) into an infinite
number of harmonic components of orders 0,1,2, ∙ ∙ ∙ , which,
by what has just been proved, are carried over into the
Fourier components of F(s, φ) of the same orders. Con-
sequently if we can prove that a harmonic distribution
function can only be carried into the density function 0 if
the distribution function itself vanishes identically, we will
have proved that f(s, φ} is harmonic of the order m in ques-