The name is absent

68 Lectures on Scientific Subjects

A necessary and sufficient condition that, to a general har-
monic continuous density function of the mth order,

F(r, θ) = F_m(f) cos mθ+G_m(f) sin mθ,
there corresponds a continuous distribution function [unique
and harmonic of the same order m by what precedes},
f(s, φ) ≈f_n(s) cos mφ+g_m(s') sin mφ

is that the integral equation for h_m(s^),

(3)             H_m(f) = h_m(r sin u)e"^mudu

where

H_m(f) = F_m(r) +iG_m(r~),
admits of a continuous solution, whose real and imaginary
coefficients will then yield f_m(s^) and g_m(s} respectively.

The proof of this italicized statement follows immediately
from (1) and what has been proved in the preceding section.

7∙ EXPLICIT SOLUTION IN THE HARMONIC CASE Wl = I

In the harmonic case m = 1 we have to consider the integral
equation

(3ι)                H₁(F) h₁(r sin u)e~"ⁱdu,
which expresses the relation between the known coefficients
F₁(F), G₁(F) of the density function F and the like coefficients
/i(ʃ), gɪ(ʃ) of the distribution function f(s, φ^) which it is
desired to find. This equation may be written

7Λ(r) =f* h(r sin w)(cos и — i sin u)du.

But the first component of the integral on the right is clearly

while the second term evidently has the value

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