The name is absent

to be continuous in its arguments. The singular distributions
hitherto mentioned are of course not of this type.

One of the simplest possible types of two-parameter dis-
tributions, aside from the trivial case f(s,φ)=l when
F(r, 0) = 2τr, is that in which /(ʃ) =0 for ʃd and f(s) =k
for ʃ > 1 (fig. 5).

Here it is readily found from (1) that the corresponding
density function is
0 for r < 1

,,/11 . ^-11∖ _c .

4τr⅛(-— sin - for r>l.

When rectilinear erasures are not allowed (First Problem)
f must be positive or zero everywhere; if we allow rectilinear
erasures after the drawing has been made (Second Problem),
the restriction that / is of one sign is abandoned; and if we
allow only a single uniform erasure (Third Problem), we are
interested in the positive solutions /(ʃ, <p) obtained when
F(r, 0) is increased by a suitable positive constant.

4. THE SYMMETRIC CASE

A very interesting special case is that in which the den-
sity function F(r, 0) depends merely on the distance from
a fixed point, say the origin O, so that F(r, 0) = F(F) with
F( — r) =F(F). This will be called the symmetric case for
obvious reasons; the case just specified is of this type. It is
natural to conjecture that in the symmetric case we may
restrict attention to two-parameter distribution functions
f(s, φ) of like symmetric type:/(r, φ) =f(s) with/(—ʃ) ≈f(s),
i.e., corresponding to lines equally distributed in all direc-
tions about the origin.

The fact that we need only consider distribution functions
of this symmetric type may be proved as follows.

Suppose first that a поп-symmetrical continuous dis-

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