23N Page 104
motor' effect. (This happens about 5 times.) This means that the motor settings are wrong, but happen to give long patches in which the Ψ's are correctly set. It is not easy to put this [illegible]ght on Colossus and the best thing to do is to give the hand cryptographer a de-chi and 'pseudo-psi' stream. This will enable him to get a 'break' and thus to set the psis correctly and reset the motor. Finally the /34 test could fail due to the machine being out of order. The easiest way of deciding the cause of failure is, as ever, to do a letter count, in this case on undifferenced plain language.
(c) Setting Ψ's when not all the Χ's are set.
If all the Χ's are set but the motor is set, then this motor can be used for more powerful Χ runs, as already pointed out. However a more powerful method is to set the Χ's and Ψ's simultaneously.
For example if Χ1, Χ2, μ61, μ37 are set the most powerful procedure is to set Ψ1 and Ψ2 and then say Χ3 and Ψ3 together (R4 p. 46).
(d) Testing the machine.
When Ψ runs are done the machine is fully extended and test runs become particularly important. The test runs done are similar to those in the case of Χ's and motorised Ψ tapes are made available for each day's keys.
23N COALESCENCE
Suppose that the limitation in Χ2 + Ψ'1. Then the Ψ1 character at any letter of the message may have an influence on the setting of Ψ1 at the next letter. This gives rise to a remarkable phenomenon known as coalescence. For example two different hypotheses about the initial setting of Ψ1 with Χ2 and basic motor settings fixed, may give rise to the same Ψ1 setting at the nth letter and all succeeding letters. The two initial settings are then said to have coalesced by the nth letter. Two theories coalesce by the nth letter if and only if they give rise to the same setting at both the nth letter and the (n+1)th letter. It is shown in (23X) that the chance of the right setting not being coalesced with a good proportion of the 42 other hypotheses after n basic motor dots is about 1.3 e-n/760
There are several ways of taking advantage of the phenomenon of coalescence. If a message is sufficiently long, there is no need to run for Ψ1 if Χ2 and the basic motor are set. We can simply assume a conventional setting (say 01) for Ψ1 and span from about 2000 to the end, and the wheel will probably be correctly set for the part of the message used. Long runs can be replaced by short runs
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