At present the Central Council Decisions only cover the extension of methods which have hunt bells and, following a request at the Council meeting in Caerleon in 1993, the Methods Committee is intending to propose suitable amendments to the Decisions at the Council meeting in Salisbury this year so that they also cover the extension of principles (methods without hunt bells).

This article describes the amendments under consideration and gives those who are interested in the subject an opportunity to comment before the motion to Council is finalised at the next meeting of the Methods Committee.

We have restricted our attention to formulae for extension which keep the length of the lead constant at all stages. This would not preclude consideration in the future of additional formulae in which the length of the lead increased at higher stages, although such formulae would possibly be of limited application.

All principles may be reversed and, unless a principle is Double, its reverse will be a distinct principle. Particularly at the Doubles stage, where the greatest number of different principles have been rung, the reverse has usually been given the same name with the prefix "Reverse". The proposition is that this practice should become a requirement and, moreover, that the extension of the reverse should be the reverse of the extension. Expressed another way, this means that if a valid extension of the reverse of a principle were rung and named then this would also define the extension of the original principle.

Experience with the extension of methods (with hunt bells) suggests that the most satisfactory extension constructions are those which are applicable at an indefinite number of higher stages, for example, the construction produces a method at alternate higher stages or at every fourth higher stage. The least satisfactory extensions are often those where the construction produces a method at just one higher stage. The proposition is that the extension construction of a principle must be applicable at an indefinite number of higher stages.

The Decision on Method Extension (Decision (G)) talks about extension constructions being either "static" or "expanding." To emphasise the symmetry of the formulae under consideration this article instead uses the equivalent expressions "static with respect to the lead" and "static with respect to the lie."

Given that the length of the lead is constant, the formula must say what happens to the places made within the lead since it is these which define the principle. We have considered the following four formulae where Formula 1 is the least restrictive and Formula 4 is the most restrictive.

Each place either remains static with respect to the lead or remains static with respect to the lie. For example, if the parent stage were Major, a place in 3rds could either remain fixed with respect to lead, that is remain in 3rds, or remain fixed with respect to lie, that is become successively 5ths, 7ths, 9ths, *et seq*.

Either all places remain static with respect to the lead except for those places contiguous to the lie or all places remain static with respect to the lie except for those places contiguous to the lead.

All places up to a particular position remain static with respect to the lead while all places above that position remain static with respect to the lie. This is called extension by "modes" where, for example, mode-5 means all places up to and including 5ths place remain static with respect to the lead while places above 5ths place remain static with respect to the lie.

Either all places remain static with respect to the lead or all places remain static with respect to the lie. This would regularise the eleven principles which have so far been rung at more than one stage but, with the exception of Original (odd and even stages), would prevent Double principles from extending to Double principles.

There are a number of additional restrictions which would probably be desirable, primarily to avoid introducing unexpected pieces of work, regardless of which formula were used:

(i) Symmetric principles remain symmetric (about the half-lead change).

(ii) External places remain external. That is, places at lead remain at lead and places at lie remain at lie.

(iii) Contiguous places in a change remain contiguous. For example, places in 3-4 either both remain static with respect to the lead or both remain static with respect to the lie.

(iv) The number of consecutive blows in the same position is neither increased nor decreased, This means, for example, that a place causing two consecutive blows in 3rds place may not remain static with respect to the lie while a place causing two consecutive blows in 5ths place remains static with respect to the lead since this would produce three consecutive blows in 5ths place in the extension.

(v) Within one change all the places remaining static with respect to the lead must be below all the places remaining static with respect to the lie. For example, 3rds place may not remain static with respect to the lie while 6ths place in the same change remains static with respect to the lead which would produce places in 5-6 at the first stage of extension (what would happen at the second stage of extension is left as an exercise for the reader!).

As part of our investigations into the implications of these formulae we produced all possible extensions of rung principles using Formula 1, subject to all the additional restrictions, and which gave a valid principle at alternate higher stages. 86 principles produced 375 extension constructions and the notations for each of these were generated at the first five stages of extension. We are very grateful to Julian Morgan who produced blue-lines of most of these principles to help us with our investigations.

Although many of these constructions have interesting blue lines, in some cases work is introduced in the extensions which does not occur in the parent. Formulae 2, 3 and 4 all represent subsets of Formula 1 and we will aim to choose the most liberal formula which minimizes the likelihood of recognising extensions which introduce unfamiliar work. We think that it would be better initially to adopt a fairly restrictive formula, which might possibly be relaxed in the future, rather than to adopt a more liberal formula and then be faced with introducing additional restrictions to prohibit unsatisfactory extensions.

If you have comments on any of these suggestions, we would like to hear from you, preferably in writing.

Incidentally, the Decision on Method Extension is available in *The Council's Decisions* from Central Council Publications (see regular advertisement) and an explanatory handout is available from the Methods Committee on receipt of an sae. Both these are also available from the Methods Committee file area on the Ringers' Bulletin Board (see *Computing Matters - 8, RW* 1994 pp.430-1).

Anthony P Smith

Chairman, Methods Committee

72 Buriton Road, Winchester,

Hampshire, SO22 6JE

Internet: tps@mfltd.co.uk

*The Ringing World*, February 17, 1995, page 183