The name is absent

72 Lectures on Scientific Subjects

⅛ΛW+∑(7'*.W cosτn0+G_m(r) sin mθ),
m≡ι

and if F_n (r) and G_m(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)   I_m(s')=^F_m(t')+iG_m(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 f_m(s), g_m(s~) defined by

⁽⁷⁾ ∕_mω+⅛_mω=gr_mω

form the Fourier coefficients of a continuous function f(s, φ)
corresponding to

OO                              ■

l/o(ʃ) +∑(∕m(∙f) cos mφ+g_n(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.

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