24C Page 121
It is not easy to find which ΔΧ1 and ΔΧ2 do make this bulge a maximum.
It is however, very easy, if one is given, to find the other, viz. by 'taking the known wheel through the rectangle' (details below).
Accordingly the method used is to find somehow a crude approximation (a start) to one wheel, say ΔΧ2, take it through the rectangle to get ΔΧ1, take this through to get a new ΔΧ2 and so on till ΔD12 = . is a maximum. The rectangle is than said to be crudely converged.
Unfortunately this maximum may be only a relative maximum (false convergence) in the sense that though the score cannot be increased by changing either wheel separately, it can be increased by changing both wheels at once [24W(c)]. For this reason the most important item in convergence is finding a correct start.
(c) To take a wheel through the rectangle place the given wheel (say ΔΧ1) against the first row of the rectangle and add all the entries therein, changing their signs wherever ΔΧ1 is a cross (and counting 0 where ΔΧ1 is 'doubted'). According to whether this sum is positive or negative, the first character of ΔΧ2 is taken to be a dot or cross. Likewise for all rows.
It is easy to see why. The rectangle entries are bulges of ΔZ12 = . and if their signs are changed where ΔΧ1 = x they become bulges of ΔZ12 + ΔΧ1 = ., i.e. of ΔD12 + ΔΧ2 = .. The sum of these for a particular row is the total ΔD12 + ΔΧ2 bulge against the corresponding character of ΔΧ2 (the 'score for this character'). By giving each character of ΔΧ2 the same sign as this bulge, each is made to contribute positively to the bulge of ΔD12 = .. With the given ΔΧ1 and this ΔΧ2 the ΔD12 bulge for the whole rectangle is the sum of the moduli of the scores for ΔΧ2 characters.
When the rectangle is converged, the bulge is, by definition, a maximum (possibly only relative). If a wheel is taken through again, the score (which will certainly not diminish) must remain constant. In other words the sum of the moduli is the same for ΔΧ1 and ΔΧ2. It is easy to see that, conversely, when two consecutive scores are equal, the rectangle is converged. This is a useful check.
It is found better not to take all characters of a wheel when
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