General Report on Tunny


22C Page 48

22C     THE MOTOR STREAM


(a) Definitions.

For a given set of wheel patterns we define

  Number of dots in μ37 as d
  Proportion of dots in μ37 as D ≡ d/37
  Proportion of crosses in μ37 as a' ≡ (1 - D)
  Proportion of crosses in TM as a
  Proportion of crosses in μ61 as k

(b) The motor wheels.

μ61 is constructed so that 30 ≤ k ≤ 50 and k ≠37 without more than so far as is known 5 consecutive dots or 15 consecutive crosses.

μ37 is constructed so that 14 ≤ d ≤ 28 without more than, so far as is known, 5 consecutive dots or 6 consecutive crosses.


(c) The basic motor.

  Theorem I. The BM has a period of 61 x 37 = 2257 (C1)

Proof          After n complete revolutions of μ61, μ37 has moved nk places. The initial position is reached when
  nk ≡ 0 (modulo 37)

Since k ≠ 37, n must be multiple of 37, and the motor returns to its original position after 37 revolutions of μ61.

  Theorem II. Proportion of crosses in BM = a' (C2)

Proof          Since the period of the BM = 2257, each position of μ37 occurs with each position of μ61 once in each cycle. As each character of μ37 occurs 61 times per cycle, the proportion of crosses in μ37 is not changed by the extension.


(d) The total motor.

Assuming that the proportion of crosses in the limitation is ½ - which is not strictly true for Χ2 or Χ2P5 limitation we have:

Proportion of dots in TM = ½ x proportion of dots in BM
  i.e. 1 - a = ½(1 - a')

Proportion of crosses in TM is comprised of:
  Proportion of


BM dot lim dot ½(1-a')
BM cross lim dot ½a'
BM cross lim cross ½a'


(C3)

Summary

  d/37 = D = 1 - a' = 2 (1 - a) (C4)

(e) Double dots in BM.

The proportion of double dots in the BM is empirically 1.1( 1-a')2 (se


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