72 Lectures on Scientific Subjects
⅛ΛW+∑(7'*.W cosτn0+Gm(r) sin mθ),
m≡ι
and if Fn (r) and Gm(r) vanish to the(m-l)th order in r for all
m,i then there will exist a corresponding continuous distribu-
tion function if and only if the integrals
(6) Im(s')=^Fm(t')+iGm(t')}
("+√^)*⅛-√≡¾, (u.⅛i),
√u≈≡l t "
are continuous for j≥0 together with their first derivatives in
such wise that the continuous functions fm(s), gm(s~) defined by
(7) ∕mω+⅛mω=grmω
form the Fourier coefficients of a continuous function f(s, φ)
corresponding to
OO ■
l/o(ʃ) +∑(∕m(∙f) cos mφ+gn(s') sin mφ).
m=ι
In this case f(s, φ) forms the unique continuous solution of the
distribution problem.
The proof is readily made by means of an induction based
on the equations of the Note appended to the present paper.5
II. ON THE THREE DRAWING PROBLEMS
On the basis of what precedes we obtain immediately the
following general conclusions for the three drawing problems
specified earlier:
(1) A drawing with given continuous6 density function
F(r, θ) ≥0 can be made without Tectilinearerasures by means
4This condition will be satisfied, for instance, if F(r, 0) has continuous partial
derivatives in x and y of all orders k in open continua S* containing the origin.
5See also my paper already referred to.
5That is, continuous in x and y.