# Turing's Treatise on the Enigma

of the L.H.W. in which these pairings occur, the U.K.W. being understood to be in the zero position. Either form of short catalogue may be made by setting up the L.H.W. rods paired according to the U.K.W. as in Fig 54, and analyzing the resulting pairs.

To understand the use of the short catalogue when the U.K.W. rotates we must remember that if the U.K.W. and L.H.W. are rotated in step the effect is a slide along the diagonal of the resulting pairs. If we are given actual pairs for which the U.K.W. was not in the zero position we can slide the pairs along the diagonal until we have pairs which would have occurred with the U.K.W. in the zero position. This will show up on the catalogue because there will be a number in common in the squares under these pairs. For instance in the case of the DANZIGVON crib we found the middle wheel to be red in position 11. This gives the middle wheel couplings pn, ve, hn, uy as consequences of the R.H.W. couplings qn, uk, fx, ep. These can be read off from Fig 55, although of course we should only set up the M.W. inverse rods in a case where we did not know the M.W. position. If we slide pr, hn, ev, uy, ten places forward along the diagonal we get wg, mi, zf, ke and in each of the squares wg, mi, zf, ke on the green (L.H.W.) short catalogue we find the number 4, i.e. these pairs occur at U.K.W. 0, L.H.W. 4: consequently qn,… occur at U.K.W. 10, L.H.W.14. The mechanical process would actually be to take pr on the small sheet of the catalogue and lay it against ve on the large sheet. This automatically results in wg and zf being together and all other pairs of pairs resulting from sliding pr, ev along the diagonal. We look in the pairs of squares to see if there are numbers in common. When we find such a case we have to look in a third square resulting from sliding hn. It is as well therefore to have rulers in gauge with the catalogue to measure off the distances. Having found the right amount of slide forward on the diagonal, i.e. to the right in the catalogue we calculate the positions of the wheels from the formulae

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