The name is absent

64 Lectures on Scientific Subjects

In the accompanying drawing (fig. 7) Mr. Middleton takes
only sixteen lines for each radial direction and lets the angle
increase successively by 5⁰. His result is shown herewith
and evidently accomplishes the desired result as far as
could be expected in view of the relatively few lines used for
each direction.

If, on the other hand, we take a good many lines for each
direction, these directions being at a considerable angle (say
τr∕4 radians) apart, another interesting type of approxima-
tion is obtained (fig. 8).

Since a circular white spot is the “negative” of a dot, it is
clear that we can draw a “faint negative” of any drawing
by a process of stippling. More precisely, we can reproduce
a doubly exposed negative obtained from a first weak ex-
posure to a stippled reproduction of the given drawing and
a subsequent strong exposure to uniform light.

6. THE GENERAL HARMONIC CASE

Imagine now a continuous distribution function /(ʃ, φ)

.         .                          .              .                           . 2τr

which is a simple harmonic function of φ of period — :

/m(ʃ, <P) =‰(j) COS τn<p+g_m(j) sin mφ.

Here∕_m(j), g_m(j) are to be regarded as even or odd according
as m is even or odd, so that the functional identity
fm( -ʃ, +0 =‰(-s^,> <p} holds and ∕_m(0) = g_ra(0) = O.

We shall speak of such an ∕_m(j, φ) as harmonic of the with
order (wι≥O); in particular, the symmetric case is the har-
monic case of zero-th order.

A density function F(r, θ) will similarly be said to be har-
monic of the with order if

F{r, θ~) =F_m(r) cos mθ+G_m(r) sin mθ.

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