22H Page 67

limitation can be derived directly from (H1) by treating the limitation as ΔΨ'_{6}. Since ΔD_{6} = ΔΨ'_{6}, ΔP_{6} must be regarded as a dot, and P(ΔP = Θ) put equal to zero, where Θ is a "letter" whose 6th impulse is a cross.

As a dechi is usually counted when Z and chis only are known, it is only possible to count against limitation dots and crosses when Χ_{2} lim. is being used.

(c) __Some ΔD counts__.

In practice it is arduous to obtain information about the frequency of letters in ΔD by means of ΔP counts and the relation (H2). The simplest way of obtaining information is by collecting ΔD counts from chi-setting messages or by combining ΔP and ΔΨ' on a Robinson or Colossus. This was not at first realised (R1, 31,79; R2 37,51)

In Figs. 22 (VI)(VII)(VIII) are shown ΔD counts corresponding to three different ΔP counts (Types A, B, C) and the ΔΨ' counts given in Fig. 22 (V). As with the ΔΨ', ΔD counts are given separately for Χ_{2} lim. and Χ_{2} Ψ'_{1} lim., and in the case of Χ_{2} lim. the counts of ΔD against L = x and L = **.** are given separately.

The counts show the gradual flattening of ΔΨ' and ΔD as the dottage decreases, and also how this flattening is to some extent marked by random variations. The importance of good T.M. x letters is shown particularly by the Type A figures. Here the ΔP (and ΔD) counts are dominated by one very powerful letter (/), with the result that 8 = (/ + 8) is the second highest letter in ΔD. The importance of 8, V, X, 5, Q, K etc. is even more marked in the counts of ΔD against L = **.** as given for Χ_{2} limitation.

(d) ΔD counts with Χ_{2}P_{5} limitation.

With Χ_{2}P_{5} limitation it is not possible (in practice) to count ΔD against lim. dot and lim. cross, but it is possible to count ΔD against Χ_{2} dot and Χ_{2} cross. (R3 p. 56).

When P consists of German language P_{5} → dot (See 22G) and therefore L = Χ_{2} + P_{5} → Χ_{2}. Therefore rather more than half the bulge on good language letters (3, U, F, J etc.) in ΔD comes against Χ_{2} crosses.

The strength of 5 in ΔP is largely derived from P = 5M89, with ΔP = UA5. Now when 5 occurs in this way in ΔP, P = 5 and P_{5} → x.

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