From the X difference patterns, it was possible to determine the correct X
it patterns with some ambiguity. Actually each assumption about the unknown
differencing interval led to a different X pattern, but most of these could be
rejected as having too many, or too few crosses. The justification for this lay in
the fact that in July and August 1941 the numbers of dots and crosses in any X or
PSI wheel patterns had been made as nearly equal as possible.
Those few possible X wheels that remained for one of the impulses were
applied in their proper settings to the alternative subtractor keys, and the
resulting sums were examined to see if they were nearly periodic. One of them
did indeed prove to be an extended PSI key.
So the ambiguity of the subtractor keys was resolved, and one impulse of
each key was successfully broken down into X and PSI keys. By studying the
PSI' key in the impulse it was possible to decide, for very many of the subtractor
letters just how many PSI movements had intervened between them and the beginning
or the message. As the PSI movement was the same for all five impulses, it
followed that for very many letters of the key, the settings of all the PSI
wheels, relative to their initial settings could be determined. This was done,
and then the value dot was assumed for the first character of the X wheel in
another impulse. This assumption was legitimate, since the patterns of both X
and PSI wheels in any impulse can be reversed without affecting their sum. Then
from the characters of the key corresponding to the first position in this X
wheel, a number of characters in the PSI pattern were obtained, and put at
their proper intervals in the PSI pattern, by the use of the relative settings.
From other key characters corresponding to these PSI characters, more X
characters were found, and then by continuing this process the complete X
pattern and PSI patterns were built up.
Hence all the X and PSI patterns were determined and then the motor key
was analysed just as for July and August, 1941.
The message setting method was then applied to the Key from the depth or
three and this was successfully set on the X and PSI wheels which had been
derived from the near depth. The motor wheels were however different.
(e) Value of a and b
When the March wheel patterns were inspected it was seen that there were
still 11 dots in Mu37 ( so that a = .703 since there was no limitation) and that the
value of b was about .7 giving ab = 1/2. These values must be compared with those
for the patterns for 1941 when a =.703 b < 1/2 so that ab was always
less than .352.
The change in the value of b explains the failure of the old method of key
analysis on the key from the depth of three. It is worth noticing that the Tunny
machine would probably never have been broken if there had been no stretch of key
susceptible to the single impulse analysis possible when ab not = 1/2.
42C MESSAGE SETTING FOR MARCH 1942
The success obtained with the near depth of March 3rd. confirmed the theory
of indicators which has been mentioned above. It was now taken for granted that
the setting of each wheel was