The Weibull Distribution: A Handbook - Index of

1.1 Genesis of the WEIBULL distribution 11
Original German text English translation
Für xmax = s wird D = 100, woraus man
weiter erhält
a = 100nb
1 − e −b sn . (8)
Hierin kann e−b sn
Dann ergibt sich
© 2009 by Taylor & Francis Group, LLC
≈ 0 gesetzt werden.
D = F(x) = 100 � 1 − e −b xn� , (9)
xmax = s leads to D = 100 and further to
a = 100nb
1 − e −b sn . (8)
In (8) one can set e−b sn ≈ 0. So the result
is
D = F(x) = 100 � 1 − e −b xn� , (9)
R = 100 − F(x) = 100e−b xn,
(10) R = 100 − F(x) = 100e−b xn,
(10)
f(x) = 100nbx n−1 e−b xn.
(11) f(x) = 100nbx n−1 e−b xn.
(11)
Die Ableitung ist also spekulativer Natur,
indem von einer der Hauptformen der
Kornverteilungskurve erfahrungsgemäß
angepaßten Gleichung ausgegangen wird
. . .
The derivation is speculative in essence
because it starts from an equation which is
fitted to the main form of the distribution
curve by experience . . .
RAMMLER continues to describe the approach made by him and ROSIN giving the same
result. Their heuristic and graphic–oriented approach is based on grinding experiments and
their observation that the passage through the mesh — expressed as a function of the finesse
of the sieve — behaved like a general parabola. After several algebraic manipulations
the equation for R is the same as (10). Further, RAMMLER (1937) reports on a paper
by J.G. BENNETT (1936) based on their paper of 1933. The decisive part that slightly
connects the WEIBULL distribution, heuristically discovered as the law governing the size
distribution of ground material, and the WEIBULL distribution, theoretically derived as the
limit distribution of the smallest sample value by FISHER and TIPPETT, reads as follows
(RAMMLER, 1937, p. 164):
14 The paper does not contain any details about this kind of reasoning. (Remark of the author)
15 D is the passage through the mesh and is thus the complement to R. (Remark of the author)
16 At this point RAMMLER cites the paper from 1933 of ROSIN, RAMMLER and SPERLING. (Remark of
the author)
17 This is a density which is skewed to the right and has a mode not on the border of the domain. (Remark
of the author)