The original of this document is held in the United Kingdom at
The National Archives, Kew, Richmond, Surrey, TW9 4DU. Reference HW 25/3.

# Turing's Treatise on the Enigma

## Chapter V. Coupling Catalogue

When we have found the rod position of the R.H.W. and a few couplings for a message it is possible to find the positions of the other wheels from a suitable catalogue.

Short catalogue

One method is to try independently all the possible positions for the middle wheel. We shall want to know the middle wheel couplings which are consequences of these various assumptions. This can be done by setting up inverse rods for the middle wheel. The rods are paired off according to R.H.W. couplings, i.e. M.W. output. This has been done for the couplings ku, fx, ep which arose in the DANZIGVON crib in Fig 55, assuming the red wheel in the middle. The pairs in each column of this set up give possible M.W. couplings. We have now to find out whether these couplings are possible. Our procedure is rather different according as the U.K.W. does or does not rotate. In the case that the U.K.W. does not rotate it will be sufficient to have a Foss sheet (the rows and columns lettered preferably with the diagonal alphabet) in which, in the RW square are entered the positions of the left hand wheel at which the RW is one of the pairs in the L.H.W. output alphabet Fig 51. This is known as the ‘short catalogue’ for this wheel. To use it in connection with the DANZIGVON crib we should take each column of the Fig 55 in turn and look up the pairs in it on the short catalogue and see if all the squares had a number in common. If we found such a case the number in the square would give the L.H.W. rod position, and the column of Fig 55 would give the M.W. position. Actually the U.K.W. rotates for our example so that we would have no success.

In the case that the U.K.W. rotates, we need essentially the same short catalogue, but we arrange it slightly differently. Instead of the lines of the catalogue corresponding to fixed output letters they correspond to fixed distances on the diagonal, (eq qwertzu) between the output letters. This may be seen from Figs 52, 53 which illustrate such a catalogue. The pairings are written above the figures giving the positions.

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