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Henry Bushby Transcription pages

VM533B94v4-p065.jpg

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65


Rule of Rings

p91
In any [insk?]. r. k. the number of
ways in which 2 any 2 bights may
be cut and have their cords rejoined
so as to increase the existing value
of Z by forming an additional Z
(or "ring") is limited to ^cannot exceed B/Z -1,
(or C/Z(n-1) -1),
A ring is formed when any part
in such a case numbered y is joined
to any other part the number of
which is y+M2n -1 or y+M2n-(1+2B)
(= y+M2n-(1+2C/n-1)) or y+M2n-3
(as in this case the complementary part
of y's bight would have to break ^come within the
first part of the rule.), provided always
that the number of cases is limited
to B/Z -1, and that the values of
the part-numerals obtained by this
formula are the first (B/Z -1) one or
more in order of arithmetical greatness.

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