The general equations to correlate the distribution of the radius r of spheres randomly dispersed in the three-dimensional space with measurements on a random test plane are
Nvo =
Nao/2rand (δn) = 2
n-1√ π[
n_??_ (
n/2)/(
n+l) _??_(
n+1/2)]Q
n+1rnfor the diameter S of circular sections of spheres; and
Nvo = Nλo/πr2Q2and λ
n=2
n+1/
n+2•Q
n+2/Q
2r
2for the length λ of chords delivered by intersection of a random test line. In the above expressions
Nvo,
Nao and
Nλo are the numbers of spheres in a unit volume, of circles on a unit surface area and of chords per unit length of a test line, respectively;
n is 0 or a positive integer;
r the arithmetical mean of
r; (δ
n) and (λ
n) the means of the n-th powers of S and A, respectively; and
Qn a quotient defined by
Qn=(
rn)/
rn The ratio of measured (δ
2)/δ
2 or (λ
2)/λ
2 is used for calculating one of the parameters of assumed theoretical distribution functions. A second parameter is then estimated from δ or λ. The method was applied to the normal pancreatic islets, and the use of chord length λ was preferred to that of diameter δ, because the error due to the failure in identifying very small islet sections was minimized in the former.
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