22B Page 46
character of the ith impulse of the U-stream at a given ciphering position.
L is used to denote the operative character of the limitation.
(b) Some further definitions.
The following symbols are generally used in Tunny-analysis and must be defined here:
U = letter preceding U
Ui = character preceding Ui
= letter preceding U
U = letter following U
= letter following U and so on.
ΔU = U + U|
Δ2U = Δ(ΔU)
ΔnU = Δ(Δn-1U)
Δ2U = U +
Δ2U = U + and so on.
Uij = Ui + Uj
= Ui + a cross
Ui x : P(Ui = x) > ½
Ui . : P(Ui = .) > ½
Ui x : P(Ui = x) = p where p > ½
Ui . : P(Ui = .) = p where p > ½
(c) Two general theorems.
|Theorem I :||Δ(U + V) = ΔU + ΔV||(A1)|
|Proof :||Δ2U = Δ(ΔU) = (U + U) + (U +) = U + = Δ2U|
|Theorem II is a special case of the general theorem:|
|ΔnU = ΔnU if and only if n = 2r. (See R5 p. 114)|
22B THE CHI-STREAM.
The chi-stream differs from a random sequence of letters in its periodicity in each impulse taken separately and in the deliberately arranged equality of dots and crosses in each impulse.