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.

-80-

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