# Turing's Treatise on the Enigma

U.K.W. position = slide forward on diagonal
L.H.W. position = number in square + slide

The Turing sheets

The short catalogue should work very well when the Umkehrwalz rotates, and there is no information connecting the position of the U.K.W. with the positions of the other wheels. In the case of a fixed U.K.W. we can often make use of an analysis of R.H.W. couplings.

The lay out of the catalogues is largely determined by the special method by which they are made, but it seems to be reasonably convenient in use. The catalogue is divided into sheets numbered 1 to 13. Each of these sheets consists of a 26x26 square with margin at top and left hand side, preferably on 1/3" gauge. One such sheet is shown in Fig 56 partly constructed. The letters and numbers in ink are the only ones concerned when the sheets are being used, the others being part of the construction, and left on to help in tracing errors. The entries 10, 18, 21 in the square in column 15 and the row with KV in the margin mean that the coupling KV occurs when the M.W. is in position 15 and L.H.W. in any of the positions 10, 18, 21. In order to find the positions at which two couplings can occur we have only to find the corresponding lines of the catalogue against one another and compare the numbers in the adjacent squares. It is fairly easy to find the right sheet because the number of the sheet gives the distance along the diagonal of the two letters of the pair, e.g. K and V are at distance 5 along the diagonal (KPYXCV) and KV occurs on sheet 5.

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