The name is absent

52 Lectures on Scientific Subjects

distribution (see fig. I).² The amount of lead Q deposited
within a circle of radius r having O as center is evidently
proportional to r: Q=kr. Hence the amount of lead dQ
in the ring between a circle of radius r and one of radius
r-∖-dr (dr, an “infinitesimal”) is dQ=kdr. But the area dA
of this ring is 2πrdr, since A =τrr² is the area of a circle of
radius r. Consequently the surface density of the lead is
given by the ratio d<Q∕dA = k∕2ιrr. Hence the density ob-
tained is inversely proportional to the distance r from the
fixed point O, as required.

There arises similarly in any such problem a fundamental
density function F, depending upon position in the plane and
corresponding to the degree of blackness of the drawing
which ranges from white through gray to black. The ex-
treme cases F = Q and F = ∞ correspond to white and black
respectively. We may think of F as measured by the depth
of the deposit of lead on the paper. Of course in actual prac-
tice not only has F a certain effective maximum, after which
the lead does not adhere to the paper, but F will change
gradually from point to point. However in our idealization
of the problem we shall not always require F to be finite and
continuous. Evidently in the special case considered above
F becomes infinite at the point O.

On the other hand we shall always assume that the
amount of lead laid down, JFdA (dA, element of area) is
finite in any finite part of the plane.

An interesting variant of the general problem of rec-
tilinear drawing specified above is obtained when we allow
rectilinear erasures to be made after the drawing has been
completed, with the natural requirement, of course, that no
lines already drawn are to be erased. Here if F_d is the

²The drawings shown in this paper have been very kindly supplied by Mr.
David Middleton, to whom I desire here to express my warm appreciation.

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