The law governing the extension of the sequence PSI was still unknown.
The four impulses of the key were next attacked, and they were successfully
broken down into X and PSI patterns, just as the first impulse had been. In these
cases the periods of the X wheels were found by booking 7-sign repeats in the
first few hundred places of each impulse, factorizing the intervals and selecting
the most common, fairly large, prime factor. The periods were found to be 41, 31,
29, 26, and 23 for X1, X2, X3, X4, and X5 respectively, and 43, 47, 51, 53 and 59
for PSI1, PSI2, PSI3, PSI4, and PSI5 respectively.
(c) The Motor
The next problem was to determine the law governing the extensions of the
PSI patterns. This was attacked by means of the concept of the "motor-key".
The motor-key was defined as a sequence of dots and crosses of which each
sign was associated with a particular pair of consecutive signs of PSI' , and such
that the nth sign of the motor key corresponded to the pair formed by the nth and
(n + 1)th signs of the extended PSI key. When two consecutive signs in PSI'
correspond to the same positions of the PSI wheel the corresponding motor-key
sign was defined to be dot, and when two such signs corresponded to different
positions of the PSI wheel the corresponding motor key sign was defined to be a
The motor key corresponding to a particular impulse could only be determined
partially from the corresponding PSI' key. When for example a block of 3
consecutive crosses in the PSI wheel was represented by a block of 5 consecutive
crosses in PSI' , it was possible to say that just two of the pairs of consecutive
crosses in this block corresponded to dots in the motor key, but it was not
possible to say which two of the four such pairs these were.
A pair of consecutive different signs in PSI' necessarily corresponded to a
cross in the motor key, but the position of a dot in the motor key could only be
fixed, when it corresponded to the extension of a singleton dot, or cross, in the
PSI pattern. As there were very few singleton dots, or crosses in the PSI
patterns, very few dots could be fixed in the motor key. Sometimes a group of
several consecutive dots, or crosses, in the PSI key would not be extended at all:
each sign in the motor key corresponding to a pair of signs in this block could
then necessarily be a cross.
A motor key determined from a PSI' key therefore consisted of a number of
isolated groups of one or more crosses, together with a few groups consisting of
dots flanked by crosses. These groups would be separated by intervals whose
lengths varied from two places up to eight or nine. In each such interval the
number of dots, but not their distribution, would be known.
A study of the indicator had suggested the hypothesis that the motor keys of
the five impulses were identical. For since the first and second indicators
affected only the first and second impulses respectively, it was supposed that
each indicator letter gave the setting of a particular wheel in the machine. We
have already mentioned the evidence that the twelfth indicator letter gave the
setting of a wheel period of 23. This wheel could now be identified with the
fifth X wheel. It seemed probable therefore that the first five indicator letters
corresponded to the five PSI wheels in order, and the last five to the five X
wheels in order
This left only the middle two indicators to govern the motor key. (This
would explain why near depths differing in this pair of indicators have proved
But five independent motor keys should need at least five