The name is absent

Rectilinear Drawing          73

of a suitable Continuous⁷ two-parameter distribution function
/(ʃ, φ) if and only ifʃ(ʃ, φ) exists as specified above and is
positive or zero. If such a solution exists it will be unique.

(2) A drawing with given density function F(r, 0) ≥0 can
be made by means of a continuous two-parameter continuous
distribution function /ɪ(ʃ, <ρ) ≥0, followed by a similar dis-
tribution function f₂(s, φ) of non-overlapping rectilinear
erasures, if and only if/(ʃ, φ) exists as specified above, when
we may take

/i(ʃ, <p) = (l/(ʃ, <P) I +/(ʃ, ^))/2, Ж v) = (l/(ʃ, <P) I -/(ʃ, ^))/2
so that

/(ʃ, <P) =√i<Λ φ) “.MA φ)∙

If such a solution exists it is unique, and evidently requires
the least possible drawing and subsequent erasure.⁸

(3) A drawing with given continuous density function
F(r, θ) ≥0 can be made by means of a continuous two-para-
meter distribution function /(ʃ, φ) followed by a single
uniform erasure if and only if/(ʃ, <p) exists as specified above.
If /(ʃ, φ) is positive or zero everywhere no subsequent
erasure is of course necessary. If, however, /(ʃ, φ) has a
negative minimum — m, we first make the drawing with
positive density function F(r, 0) +2τrm and corresponding
distribution function /(ʃ, ^)÷w≥0, and then make a uni-
form erasure with density 2τrm. Evidently this solution is
essentially unique, and requires the least possible uniform
erasure.

It is interesting to note that in the first problem when no
erasures are allowed, certain obvious geometrical conditions
must be satisfied if the drawing is to be possible. In order to

⁷That is, continuous in s and φ.

⁸I.e., as measured by lead put down and lead erased.

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