1. Introduction

The history of mathematics records several occasions when failure of the prevailing symbolic language to cope with unwieldy expressions has led to the development of a more compact notation by which complicated ideas could be expressed intelligibly. Such innovations have in turn led to an increase in knowledge, for the chief merit of a compact notation is that the significance of statements can be readily grasped and their implications more easily studied.

With this experience as a guide, a symbolic approach to the theoretical aspects of change ringing promises to be attended by similar advantages.

The present paper records an attempt at applying such a treatment to the properties of false course heads in treble-dominated Major methods. Many of the results obtained are well known, but others have not, as far as the writer is aware, been previously published.

2. Symbols

Before proceeding to applications of the symbolic treatment, the symbols employed must be defined and their properties demonstrated. As there is a superficial resemblance between the behaviour of the symbols and the rules of ordinary algebra, it will be natural to pursue the analogy by borrowing from algebra any technical terms required.

Capital letters of the italic (sloping) alphabet will be used to denote rows. Certain letters will be given a permanent significance, while others will be defined as occasion demands, or will represent unknown rows (in the same way that *x* proverbially represents the unknown quantity in algebra).

Ordinary numbers will be represented by small (lower case) italic letters.

The multiplication sign ‘×’ will be used to mean ‘transposed by.’ For example, 4372856 × 8627534 = 6845237. When rows are represented by letters, however, the multiplication sign will usually be omitted (as in algebra) and *U* transposed by *V* will be written *‘UV.’*

The multiplication of rows differs in an important respect from ordinary multiplication. 5 × 2 = 2 × 5, and *yx* = *xy*; but *VU* is not generally equal to *UV*. Symbols must accordingly be written in the correct order.

4. Division

Transposition of one row by another having been regarded as multiplication, we will next consider what operation on rows corresponds to division. Now, in algebra, *x* ÷ *y* is often written *xy* ^{-1} that is *‘x* divided by *y’* is the same as *‘x* multiplied by the reciprocal of *y.’* The operation of division has thus been converted into one of multiplication.

Applying the same rule to rows, *‘U* divided by *V’* will be translated into *‘U* multiplied by the reciprocal of *V’* and will be written *‘UV* ^{-1}*,’* where the notation *‘V* ^{-1}*’* indicates the reciprocal of *V*, as well be explained in a later section.

The reciprocal of a given row is the row by which the given row must be transposed to produce rounds.

Taking the same values for *U* and *V* as in Section 3,

U ÷ V = | 4372856 | ÷ 8627534 |

that is, UV ^{-1} = | 4372856 | × (8627534)^{-1} |

= | 4372856 | × 4786352 |

= | 7568324 | |

In practice, division can be performed directly by writing the row to be divided over the divisor. The quotient is then obtained by writing down the numerals in the top row in the positions indicated by the numerals under them.

Example: | ||

U = | 4372856 | As 7 is over 2, write 7 in second place; as 5 is over 3, write 5 in third place; and so on. |

V = | 8627534 | |

UV ^{-1} = | 7568324 | |

Incidentally, taking the reciprocal of a row is merely a special case of division, with rounds as the numerator.

Example:2345678 | |

V = | 8627534 |

V ^{-1} = | 4786352 |

In algebra, if a term *a* is multiplied by itself, the result is written *a* ^{2} similarly, *a* × *a* × *a* is written *a* ^{3}. The small numeral is called the ‘index’ and indicates the number of *a’*s to be multiplied together. *a* ^{2} is is second power of *a* and is read as *‘a* squared’; *a* ^{3} is the third power and is called *‘a* cubed’; the fourth power, *a* ^{4}, is read as *‘a* to the fourth’ and similarly with other high powers.

(The first power, *a* ^{1}, is *a* itself, and the index ‘^{1}’ is written only when we wish to emphasise that we are thinking of *a* as its own first power.)

The same notation will be applied to rows.

Let T = | 4573862 | |

T ^{2} = | 4573862 | × 4573862 |

= | 7365284 | |

T ^{3} = | T ^{2} × T | |

i.e. T ^{3} = | 7365284 | × 4573862 |

= | 6583427 | |

If a number of powers of the same term are multiplied together, the index of the result is the sum of the indices of the powers. For instance, *T* ^{2} × *T* ^{3} = *T* ^{2+3} = *T* ^{5}; and *T* ^{4} × *T* ^{2} × *T* = *T* ^{4+2+1} = *T* ^{7}. To realise this, it is simply necessary to remember that *T* ^{2} = *T* × *T* and *T* ^{3} = *T* × *T* × *T*, therefore *T* ^{2} × *T* ^{3} = *T* × *T* × *T* × *T* × *T* = *T* ^{5}.

We are now in a position to show, as promised in the preceding section, that the index ‘^{-1}’ means ‘the reciprocal of.’

Still taking *T* = 4573862, its reciprocal is 8523746.

Now T ^{3} × reciprocal of T | ||

= | 6583427 | × 8523746 |

= | 7365284 | |

But 7365284 = | T ^{2} | |

that is, T ^{3} × reciprocal of T = T ^{2} |

In order that the rule regarding the multiplication of powers shall apply (as it must if it is to be a general rule) the reciprocal of *T* must be written *‘T* ^{-1}.’

A meaning can also be found for a zero index.

For *T* ^{1} × *T* ^{-1} = *T* ^{1-1} = *T* ^{0}

But *T* ^{-1} is the reciprocal of *T* and, by definition, any row transposed by its reciprocal produces rounds.

Similarly, the zero power of any row is rounds.

If we have to evaluate a power of a term which is itself a power, such as (*T* ^{2})^{3}, the index of the result is the product of the indices, in this case *T* ^{6}, since (*T* ^{2})^{3} = *T* ^{2} × *T* ^{2} × *T* ^{2}.

If we have an expression such as *X* = *UV*, we may wish to express *X* ^{-1} also in terms of *U* and *V*.

Generally, to take the reciprocal of a product, take the reciprocal of the individual factors, but write them down in reverse order.

The rows which, occur as the lead heads and lead ends of the plain course have properties not shared by rows in general.

Let H = | 3527486 |

I = | 2345678 |

J = | 2436587 |

*H*, *I*, *J* will always have this significance and will be used in future without further definition. They are strictly eight-bell rows, but as the treble is at home it is not written.

The seven lead heads and the seven lead ends of the plain course can be denoted by expressions involving only *H*, *I*, *J*.

For instance, the first lead head of the plain course of any method is *I*, which can also be written *H* ^{0} (see *H*, which can be written *H* ^{1}; the third is *H* ^{2}. Each lead head is *H* times the preceding one, hence the lead head after *H* ^{6} is *H* ^{7}. But *H* ^{7} is the seventh lead head (4263857) and the next is rounds; therefore *H* ^{7} = *I* = *H* ^{0}.

Similarly, *H* ^{8} = *H* ^{1}, *H* ^{9} = *H* ^{2}, and, in general, *H* ^{r} = *H* ^{r+7t} where *t* is any whole number, positive or negative. Thus *H* ^{r} = *H* ^{r+7} = *H* ^{r+14} = *H* ^{r-7} = *H* ^{r-21} and so on. The equivalent expression for *H* ^{r} that we shall come across most often is *H* ^{r-7}. For example, *H* ^{3} = *H* ^{3-7} = *H* ^{-4}. Conversely, *H* ^{-5} = *H* ^{7-5} = *H* ^{2}.

Since the plain course lead heads of all methods are the same, though occurring in different orders, they can always be expressed by *H* ^{0}, *H* ^{1}, *H* ^{2} … *H* ^{6}. If the second lead head of a method is *H* ^{4}, each lead head will be *H* ^{4} times the preceding one. The third lead head will thus be *H* ^{4} × *H* ^{4} = *H* ^{8} = *H* ^{1}; the fourth will be *H* ^{1} × *H* ^{4} = *H* ^{5}, and so on.

The lead ends of the plain course can be expressed in the form *JH* ^{P}. For instance, the first lead end of Plain Bob is 3254768 = *JH* ^{6}, the second is 5372846 = *JH* ^{5}; the last is *JH* ^{0} = *J*.

These expressions have the property that each is its own reciprocal; for example, (6847253)^{-1} = 6847253. In general, *JH* ^{P} = (*JH* ^{P})^{-1} = *H* ^{-P}*J* ^{-1} = *H* ^{-P}*J* (since *J* ^{-1} = *J*).

If the first lead end of a method is *JH* ^{b}, the other lead ends of the plain course are each *JH* ^{b} times the preceding lead head. For example, if the second lead head is *H* ^{a}, the second lead end is *H* ^{a}.*JH* ^{b} = *JH* ^{-a}*H* ^{b} = *JH* ^{b-a}. The third lead end will be *H* ^{2a}*JH* ^{b} = *JH* ^{b-2a}.

Conversely, if a given lead end is *JH* ^{c}, we can find the head of the lead to which it belongs. From above, we have:

A lead end *JH* ^{b-a} follows a lead head *H* ^{a}.

As c takes the place of b-a, we have | c = | b-a |

c-b = | -a | |

a = | b-c | |

Hence a lead end *JH* ^{c} follows a lead head *H* ^{b-c}.

The lead heads and ends of a course other than the plain course are represented by *GH* ^{0},*GH* ^{1},*GH* ^{2} … *GH* ^{6} and *GJH* ^{0},*GJH* ^{1} … *GJH* ^{6}, where *G* is the course head.

8. False Course Heads of a Method

We are now in a position to derive general expressions for the false course heads of a Treble Bob Major method in respect of courses having the tenors together throughout.

A false course head occurs when two rows in the course have the treble and tenors (1,7,8) in the same places and the rows are of the same nature, both even or both odd. For under these conditions, the same row can appear in different courses.

The plain course of a method is made up of fourteen half leads, seven following the lead heads and seven preceding the lead ends. If the latter seven are inverted, they are identical in structure with the first seven. The course can thus be dissected into fourteen half leads, seven headed by *H* ^{0},*H* ^{1} … *H* ^{6} and seven headed by *JH* ^{0},*JH* ^{1} … *JH* ^{6}. For brevity, half leads headed by even rows (such as *H* ^{4}) will be referred to as even half leads, and those headed by odd rows (such as *JH* ^{2}) as odd half leads.

Different formulæ are obtained for the false course heads, depending on whether the two rows with the treble in the same place in the same half lead are of the same or opposite nature. We will deal with the last-mentioned case first.

LetThe *a*th rows of even half leads will be even; the *b*th rows of odd half leads will also be even, and one of them might have 7,8 in the same places as one of the even *a*th rows. If so, the positions of the working bells (2,3,4,5,6) will be different in the two rows, but it will be possible to select two other courses with the working bells so placed that:

(i) in one such course, the *b*th row of an odd half lead is identical with the *a*th row of an even half lead in the plain course;

(ii) in the other such course, the *a*th row of an even half lead is identical with the *b*th row of an odd half lead in the plain course.

The two courses will thus be false against the plain course and their heads will be false course heads of the method.

LetThe *b*th row of an odd half lead in a course *F* is *FJH* ^{n}*B*, giving, for case (i):

In case (ii), the *a*th row of an even half lead in a course *F* is *F*_{2}*H* ^{m}*A*, giving:

Now the odd half leads (headed *JH* ^{0} … *JH* ^{6}) differ from the even half leads (headed by *H* ^{0} … *H* ^{6}), so far as 7,8 are concerned, in that 8 occupies the positions of 7, and 7 occupies the positions of 8. For example, the half leads headed *H* ^{0} and *JH* ^{0} have the tenors in the same places, row for row, but interchanged, since *H* ^{0} = 2345678 and *JH* ^{0} = 2436587.

Therefore if, as assumed, the rows *H* ^{m}*A* and *JH* ^{n}*B* have the tenors in the same positions, the rows *J*(*H* ^{m}*A*) and *J*(*JH* ^{n}*B*) will also have the tenors in the same positions. (*J*.*JH* ^{n}*B* = *H* ^{n}*B* since *J* ^{2} = *I*.)

*JH* ^{m}*A* and *H* ^{n}*B* are odd rows and give rise to two more false course heads, *F*_{3}, *F*_{4}, such that:

The expressions for the set of false course heads *F*, *F*_{2}, *F*_{3}, *F*_{4}, will now be examined to see whether a simple relationship exists between them.

Any false course head *F* is thus always accompanied by three other false course heads *F* ^{-1}, *JFJ*, *JF* ^{-1}*J*. These false course heads may or may not be all different and, as we shall see later, they may be accompanied by others.

9. False Course Heads Due to Rows of Like Nature

The preceding section dealt with the case where the two rows with the treble in a given place in the first half lead of a method are of opposite nature. The following formulæ are derived for the case where the two rows are of the same nature.

LetAs the rows *C*_{1}, *C*_{2} are of the same nature, even half leads can repeat among themselves but cannot repeat with odd half leads. Odd half leads can also repeat with one another, since if *C*_{1}, *C*_{2} are even (or odd), the rows corresponding to them in odd half leads will be odd (or even).

Using the same procedure as in Section 8, but noting that *J* will appear on both sides of the equations or not at all, the four possibilities for repetition are:

The relationship between these false course heads is the same as for those found in Section 8.

For10. Practical Formulæ for False Course Heads

The formulæ obtained in sections 8 and 9 are not in the form best suited for working out the false course heads of a method in practice. Some more useful expressions will now be derived from them, with the help of the following identity:If the values of H ^{0}H ^{1} … H ^{6} are written one under the other, as in the annexed list, it is found that each column of figures, read upwards, gives the coursing order of the plain course. For instance, the bells in second place, from bottom to top, are 4687532; those in third place are 2468753; and so on |
H ^{0} | = | 2345678 |

H ^{1} | = | 3527486 | |

H ^{2} | = | 5738264 | |

H ^{3} | = | 7856342 | |

H ^{4} | = | 8674523 | |

H ^{5} | = | 6482735 | |

H ^{6} | = | 4263857 |

In other words, a row *H* ^{μ} can be obtained from rounds by substituting for each bell in rounds the bell μ places earlier in plain course coursing order.

It now remains to be shown that *I*φ^{μ}*X* = *X*φ^{μ}.

In the operation of multiplying a row *W* by another row *X*, each bell in the row *W* takes up a new position (which may or may not be different from its original position). The new position is such that the bell in *n*th place in *W* now occupies the position of the bell *n* in *X*.

where *a* … *g* represent the individual numerals constituting the row *W*.

Suppose *a* ^{1} is the bell μ places earlier than *a* in plain course coursing order, *b* ^{1} the bell μ places earlier than *b*, and so on,

These formulæ are convenient for practical use, as the starting point in the calculation is the pair of rows being tested.

Similar formulæ for the three false course heads associated with *F* can be obtained by using the same transformation, but they are unnecessary, since the associated false course heads can be more easily obtained from the relationships derived at the end of Sections 8 and 9.

In applying the formulæ, a value of *n* is chosen to bring behind a pair of bells which course in the plain course. The appropriate value of in is then chosen to convert these bells into 7.8, so producing a row of course head form.

The incidence of falseness is indicated by the values of *m* and *n* so obtained. Referring to Section 8 for the case dealing with rows of opposite nature, we see that the false course head *F* involves repetition between a row in the plain course half lead headed *H* ^{m} and the half lead headed *FJH* ^{n} of the course *F*. This can be written *F*.*JH* ^{n} v. *I*.*H* ^{m}, where ‘v.’ (versus) indicates ‘false against.’

The associated false course head *F* ^{-1} is produced by repetition between the plain course half lead headed *J**H* ^{n} and the half lead headed *F* ^{-1}*H* ^{m} of a course *F* ^{-1}; that is *F* ^{-1}.*H* ^{m} v. *I*.*JH* ^{n}. The other false course heads *JFJ* and *JF* ^{-1}*J* give *JFJ*.*H* ^{n} v. *I*.*JH* ^{m} and *JF* ^{-1}*J*.*JH* ^{m} v. *I*.*H* ^{n}.

Collecting these expressions, together with corresponding expressions from Section 9, we can tabulate the incidence of falseness in terms of the half lead heads.

Table IA | Table IB | ||||

Falseness due to rows of opposite nature | Falseness due to rows of same nature. | ||||

F.JH ^{n} | v. | I.H ^{m} | F.H ^{n} | v. | I.H ^{m} |

F ^{-1}.H ^{m} | v. | I.JH ^{n} | F ^{-1}.H ^{m} | v. | I.H ^{n} |

JFJ.H ^{n} | v. | I.JH ^{m} | JFJ.JH ^{n} | v. | I.JH ^{m} |

JF ^{-1}J.JH ^{m} | v. | I.H ^{n} | JF ^{-1}J.JH ^{m} | v. | I.JH ^{n} |

In dealing with a particular method, however, we want to know which complete leads are false against one another, and we want to quote them by their position in the course, as is customary, or (better still, perhaps) by the calling positions of the tenor. The latter alternative will be adopted here, as it seems more logical to designate, for example, the Middle lead by M than to call it 1 in some methods and 3 in others.

Moreover, in a pair of methods differing only in the place made at the treble full lead, identical leads are differently numbered so that the incidence of falseness has to be listed separately for each method. When the leads are designated by the positions of the tenor, one list of falseness applies to both methods.

Now, if the first lead end of the method under consideration is *JH* ^{b}, the head of the lead to which a half lead headed *JHC* belongs is *H* ^{b-c}, as pointed out in Section 7.

We can thus rewrite Table I. in a form showing the incidence of falseness in terms of lead heads only, giving Table II.

Table IIA | Table IIB | ||||

Falseness due to rows of opposite nature | Falseness due to rows of same nature | ||||

F.H ^{b-n} | v. | I.H ^{m} | F.H | v. | IH ^{m} |

F ^{-1}.H ^{m} | v. | I.H ^{b-n} | F ^{-1}.H ^{m} | v. | IH ^{n} |

JFJ.H ^{n} | v. | I.H ^{b-m} | JFJ.H ^{b-n} | v. | I.H ^{b-m} |

JF ^{-1}J.H ^{b-m} | v. | I.H ^{n} | JF ^{-1}J.H ^{b-m} | v. | I.H ^{b-n} |

Finally, by means of Table III., we can convert the results into calling positions.

Table III. | |

H ^{0} = H | (Home) |

H ^{1} = W | (Wrong) |

H ^{2} = V | (Fifths) |

H ^{3} = B | (Before) |

H ^{4} = I | (In) |

H ^{5} = F | (Fourths) |

H ^{6} = M | (Middle) |

12. Possible Values of *AB* ^{-1}*J* and *C*_{1}*C*_{2}^{-1}

Having arrived at the stage where, given a pair of rows, we can derive the false course heads produced by them, it will be instructive to reverse the process and see what range of values *AB* ^{-1}*J* and *C*_{1}*C*_{2}^{-1} can have for a given set of false course heads.

Dealing first with rows *C*_{1},*C*_{2}, of like nature, we can rearrange the formula for *F* in the following manner:

Treating the formula pertaining to rows of opposite nature in the same way, we get:

These expressions for *AB* ^{-1}*J* and *C*_{1}*C*_{2}^{-1} are similar, the only difference being the minus sign in *H* ^{-n} in the second one, which, as will be seen, has no effect on the range of values which the expressions can take.

For brevity, *C*_{1}*C*_{2}^{-1} will be referred to as a ratio, since it is simply one row divided by another; the term ratio will, in this Section, be regarded as including *AB* ^{-1}*J* also, since, although this expression is not strictly a ratio (it is in fact a ratio multiplied by *J*) the possible values of *AB* ^{-1}*J* are the same as the possible values of the ratio *C*_{1}*C*_{2}^{-1}

Now, *m* and *n* can each take seven values (0 to 6), giving forty-nine values of the ratio for a given value of *F*. Both the expressions have the same forty-nine values, as *H* ^{-n} in one expression gives the same result as *H* ^{n} in the other when all seven values of *n* are substituted. It will thus be sufficient to consider one expression, and *F*φ^{-m}*H* ^{n} will be used.

Some practical examples will help to make clear the implications of the expression, and we will consider first the false course head set, 34562, 62345, 46325, 54263.

Table IVa shows the forty-nine values of the ratio calculated from the above expression for the case when *F* = 34562. All the ratios produced by the same value of in have been placed in the same column, while those produced by the same value of *n* are in the same row. Thus the first column of the table is obtained by putting *m* = 0, and giving *n* the values 0, 1, 2 … 6 in succession; the second column in obtained by putting *m* = 1, and so on.

Table IVa. Ratios Producing 34562

3456278 4637582 6748325 7862453 8275634 2583746 5324867 |
2638457 6825374 8567243 5784632 7453826 4372568 3246785 |
4827635 8743256 7385462 3576824 5632748 6254387 2468573 |
6745823 7562438 5273684 2358746 3824567 8436275 4687352 |

8563742 5384627 3452876 4237568 2746385 7628453 6875234 |
7382564 3276845 2634758 6425387 4568273 5847632 8753426 |
5274386 2458763 4826537 8643275 6387452 3765824 7532648 |

Now, in a set of false course heads, any course head can be regarded as *F*, since whichever is so regarded, the associated false course heads *F* ^{-1}, *JFJ*, *JF* ^{-1}*J*, are the remaining members of the set. Accordingly, by substituting 62345, 46325, and 54263 in the formula in turn, additional ratios are obtained which lead to the same set of false course heads.

Table IVb. Ratios Producing 62345

6234578 2467385 4728653 7845236 8573462 5386724 3652847 |
8426357 4685273 6547832 5763428 7352684 3278546 2834765 |
7648235 6873452 8365724 3582647 8234876 2457368 4726583 |
5867423 8752634 7283546 2374865 3426758 4635287 6548372 |

3785642 7534826 5472368 4256783 2648537 6823475 8367254 |
2573864 5326748 3654287 6438572 4867325 8742653 7285436 |
4352786 3248567 2836475 8627354 6785243 7564832 5473628 |

Table IVc. Ratios Producing 46325

4632578 6247385 2768453 7825634 8573246 5384726 3452867 |
6824357 8465273 4587632 5743826 7352468 3276584 2638745 |
8746235 7683452 6275824 3572849 5234687 2458376 4827563 |
7568423 5872634 8253746 2384567 3426875 4637258 6745382 |

5387642 3754826 7432568 4276385 2648753 6825437 8563274 |
3275864 2536748 5624387 6458273 4867532 8743625 7382456 |
2453786 4328567 3846275 8637452 6785324 7562843 5274638 |

Table IVd. Ratios Producing 54263

5426378 4657283 6748532 7863425 8372645 3284756 2435867 |
3648257 6835472 8567324 5782643 7254836 2473568 4326785 |
2867435 8723654 7385246 3574862 5436728 4652387 6248573 |
4785623 7542836 5273468 2356784 3628547 6834275 8467352 |

6573842 5364728 3452687 4238576 2847365 8726435 7683254 |
8352764 3286547 2634875 6427358 4765283 7548632 5873426 |
7234586 2478365 4826753 8645237 6583472 5367824 3752648 |

The four tables contain all values of the ratio which lead to the set of false course heads on which the tables are based.

Such a set of tables can be written out for every set of false course heads, but we have not the space to give them all here.

We will therefore confine ourselves to one or two further examples, and will next deal with the frequently occurring false course head set 32546, 46253. Table V gives the forty-nine values of the ratio leading to 32546.

Table V. Ratios Producing 32546

3254678 2437586 4728365 7846253 8675432 6583724 5362847 |
2436857 4625378 6547283 5768432 7853624 8372546 3284765 |
4628735 6843257 8365472 3587624 5732846 7254368 2476583 |
6847523 8762435 7283654 2375846 3524768 5436287 4658372 |

8765342 7584623 5472836 4253768 2346587 3628475 6837254 |
7583264 5476842 3654728 6432587 4268375 2847653 8725437 |
5372486 3258764 2836547 8624375 6487253 4765832 7543628 |

Owing to the way in which the tables are built up, a row of course head form (i.e. with 8 at home) occurs in each column of the tables. In Tables IVa, b, c, d. apart from the course head at the top of column 1 (on which the table is based), the course heads have the tenors parted. When we examine Table V, however, we discover that the course head in column 2 also has 7,8 at home and is, in fact, 46253, the remaining member of the course head set under consideration. It is clear, therefore, that a table based on 46253 would contain the same values of the ratio as Table V.

The presence in the same table of the two course heads 32546, 46253 has an effect on the incidence of falseness in a method having these false course heads. Suppose, for example, that a method has a pair of rows with the treble in the same place such that *AB* ^{-1}*J* = 3254678. The method then has a false course head *F* = 32546, with *m* = 0, *n* = 0. But it is clear that a method having *AB* ^{-1}*J* = 3254678 will also have a false course head *F* = 46253, with *m* = ^{-1}, *n* = 1, since in Table V, 4625378 is one column to the right of 3254678 and one row lower.

Similarly, whatever the values of *m* and *n* for the false course head 32546, the false course head 46253 will also occur and will have values of *m* and *n* one less and one greater, respectively, than those for 32546.

When, however, the rows concerned are of the same nature, the values of *m* and *n* for 46253 are both one less than for 32546.

The difference between the cases for rows of like and unlike nature is due to the minus sign before the *n* in the formula *F* = *C*_{1}*C*_{2}^{-1}Hnφ^{m} for rows of like nature.

An instance of such a duplication of a set of false course heads is given by the rows with the treble in third place in Cambridge Surprise, worked out in Section 11.

In this example, the additional false course head 46253 is a member of the same set as the first false course head 32546, since when *F* = 32546, *JFJ* (and *JF* ^{-1}*J*) = 46253; conversely, when *F* = 46253, *JFJ* = 32546. This turns out to be merely incidental and is not a general rule.

By writing out tables such as Table V. for all the possible false course heads, those which occur in the same table, and which thus occur together as the false course heads of methods, can be discovered. A less laborious way of finding the course heads which belong to one another is, however, available.

The first step in the procedure for working out the false course head for a given value of *AB* ^{-1}*J* or *C*_{1}*C*_{2}^{-1} consists in bringing behind a pair of bells which course in the plain course. If the ratio has more than one such pair of bells, each such pair produces a false course head, and these false course heads must appear in the same table. We can thus determine which false course heads are inseparable by examining the sixty course heads for coursing pairs.

As an example, consider the false course head 26543. Now *F* ^{-1}, *JFJ*, and *JF* ^{-1}*J* are also 26543, so that it seems as if this false course head can occur alone.

But the row 2654378 has 3,5 and 6,4 (in addition to 7,8) in plain course positions, and when we work out the false course heads produced by bringing 35 and 64 behind, we obtain 36245 and 42563 as false course heads which inevitably accompany 26543.

ForF_{1} = | 2654378 |

F_{1}H ^{2} = | 4768235 |

F_{1}H ^{2}φ^{2} = | 3624578 = F_{2} |

F_{1} = | 2654378 |

F_{1}H ^{5} = | 3582764 |

F_{1}H ^{5}φ^{5} = | 4256378 = F_{3} |

We thus see that if we have: | |||

F_{1} = 26543, m = 0, n = 0 | ) | for rows of unlike nature | |

we also have | ) | ||

F_{2} = 36245, m = 2, n = 2 | ) | ||

and | ) | ||

F_{1} = 42563, m = 5, n = 5 | ) | ||

Or | |||

F_{1} = 26543, m = 0, n = 0 | ) | for rows of like nature | |

F_{2} = 36245, m = 2, n = -2 | ) | ||

F_{3} = 42563. m = 5, n = -5 | ) |

Generally, the complete list of falseness when 26543 occurs as one false course head will be given by substituting in the relevant half of Table II. (Section 10) the following values:

F = | 26543 m | n | unlike rows |

36245 m+2, | n+2 | ||

42563 m+5, | n+5 | ||

26543 m | n | like rows | |

36245 m+2, | n-2 | ||

42563 m+5, | n-5 |

Example:

Suppose *F* = 26543, *m* = 2, *n* = 5, *b* = 1, and the rows producing this false course head are of opposite nature. Then the incidence of falseness is as follows:

F.H ^{b-n} v. I.H ^{m} | = 26543 H ^{3} v. I.H ^{2} |

= 26543 B v. V | |

F ^{-1}.H ^{m} v. I.H ^{b-n} | = 26543 H ^{2} v. I.H ^{3} |

= 26543 V v. B | |

JFJ.H ^{n} v. I.H ^{b-m} | = 26543 H ^{5} v. IH ^{6} |

= 26543 F v. M | |

JF ^{-1}J.H ^{b-m} v. I.H ^{n} | = 26543 H ^{6} v. I.H ^{5} |

= 26543 M v. F |

and *F* = 36245, *m* = 4, *n* = 7 = 0, *b* = 1.

F.H ^{b-n} v. I.H ^{m} | = 36245 H ^{1} v. I.H ^{4} |

= 36245 W v. I | |

F ^{-1}.H ^{m} v. I.H ^{b-n} | = 42563 H ^{4} v. I.H ^{1} |

= 42563 I v. W | |

JFJ.H ^{n} v. I.H ^{b-m} | = 42563 H ^{0} v. I.H ^{6} |

= 42563 H v. M | |

JF ^{-1}J.H ^{b-m} v. I.H ^{n} | = 36245 H ^{6} v. I.H ^{0} |

= 36245 M v. H |

and *F* = 42563, *m* = 7 = 0, *n* = 10 = 3, *b* = 1.

F.H ^{b-n} v. I.H ^{m} | = 42563 H ^{5} v. I.H ^{0} |

= 42563 F v. H | |

F ^{-1}.H ^{m} v. I.H ^{b-n} | = 36245 H ^{0} v. I.H ^{5} |

= 36245 H v. F | |

JFJ.H ^{n} v. I.H ^{b-m} | = 36245 H ^{3} v. I.H ^{1} |

= 36245 B v. W | |

JF ^{-1}J.H ^{b-m} v. I.H ^{n} | = 42563 H ^{1} v. I.H ^{3} |

= 42563 W v. B |

13. Groups of False Course Heads

By carrying out the procedure outlined in the preceding Section, we can marshal the sixty course heads into groups, such that if any member of a group appears as the false course head of a method, the other members of the group also occur.

If we define a set of false course heads as the four represented by *F*, *F* ^{-1}, *JFJ*, *JF* ^{-1}*J* (which may be all alike, all different, or two pairs alike), then we find that some of our groups contain only one set, some two different sets, and one three different sets. In the following list of groups, the sets within the groups are indicated by suffix numbers in the symbols *F*, *JFJ*, etc.

*Group I.* (See Note 3)

*F* = *F* ^{-1} = *JFJ* = *JF* ^{-1}*J* = 23456

*Group II.*

*F* = *F* ^{-1} = *JFJ* = *JF* ^{-1}*J* = 24365

*Group III.*

*F* = *F* ^{-1} = *JFJ* = *JF* ^{-1}*J* = 25634

*Group IV.*

F_{1} = F_{1}^{-1} = JF_{2}J = JF_{2}^{-1}J = 32546 |

F_{2} = F_{2}^{-1} = JF_{1}J = JF_{1}^{-1}J = 46253 |

F = | F ^{-1} | = 32465 |

JFJ = | JF ^{-1}J | = 43265 |

F_{1} = F_{1}^{-1} = JF_{2}J = JF_{2}^{-1}J = 32654 |

F_{2} = F_{2}^{-1} = JF_{1}J = JF_{1}^{-1}J = 45236 |

F = | F ^{-1} | = 56423 |

JFJ = | JF ^{-1}J | = 63542 |

F = | JF ^{-1}J | = 53462 |

F ^{-1} = | JFJ | = 63425 |

F = | JF ^{-1}J | = 54632 |

F ^{-1} = | JFJ | = 65324 |

F = | F ^{-1} | = 53624 |

JFJ = | JF ^{-1}J | = 65432 |

F_{1} = | F_{1}^{-1} | = JF_{1}J | = JF_{1}^{-1}J | = 26543 |

F_{2} = | F_{3}^{-1} | = JF_{2}^{-1}J | = JF_{3}J | = 36245 |

F_{2}^{-1} = | F_{3} | = JF_{2}J | = JF_{3}^{-1}J | = 42563 |

F_{1} = | JF_{2}^{-1}J | = 23564 |

F_{1}^{-1} = | JF_{2}J | = 23645 |

F_{2}^{-1} = | JF_{1}J | = 26435 |

F_{2} = | JF_{1}^{-1}J | = 25463 |

F = 34562 |

F ^{-1} = 62345 |

JFJ = 46325 |

JF ^{-1}J = 54263 |

F = 52643 |

F ^{-1} = 36524 |

JFJ = 65243 |

JF ^{-1}J = 46532 |

F_{1} = | F_{1}^{-1} | = JF_{2}J | = JF_{2}^{-1}J | = 54326 |

JF_{1}J = | JF_{1}^{-1}J | = F_{2} | = F_{2}^{-1} | = 64352 |

F_{3} = | JF_{3}^{-1}J | = F_{4}^{-1} | = JF_{4}J | = 64523 |

F_{3}^{-1} = | JF_{3}J | = F_{4} | = JF_{4}^{-1}J | = 56342 |

F_{1} = | JF_{2}J | = 45623 |

F_{1}^{-1} = | JF_{2}^{-1}J | = 56234 |

JF_{1}J = | F_{2} | = 35642 |

JF_{1}^{-1}J = | F_{2}^{-1} | = 62534 |

F_{1} = | JF_{2}J | = 52364 |

F_{1}^{-1} = | JF_{2}^{-1}J | = 34625 |

JF_{1}J = | F_{2} | = 64235 |

JF_{2}^{-1}J = | F_{2}^{-1} | = 45362 |

F_{1} = | 24536 | F_{2} = | 62453 | |

F_{1}^{-1} = | 25346 | F_{2}^{-1} = | 36452 | |

JF_{1}J = | 26354 | JF_{2}J = | 53246 | |

JF_{1}^{-1}J = | 24653 | JF_{2}^{-1}J = | 43526 |

F_{1} = | JF_{1}^{-1}J | = 34256 |

F_{1}^{-1} = | JF_{1}^{-1}J | = 42356 |

F_{2} = | JF_{2}^{-1}J | = 63254 |

F_{2}^{-1} = | JF_{3}J | = 43652 |

JF_{2}J = | F_{3}^{-1} | = 52436 |

JF_{2}^{-1}J = | F_{3} | = 35426 |

F_{4} = | JF_{4}^{-1}J | = 42635 |

F_{4}^{-1} = | JF_{4}J | = 35264 |

*Note* 2: No attempt has been made to indicate the relationship between the values of *m* and *n* for the different sets of false course heads in a group containing more than one set. A more voluminous tabulation would be necessary to show this, and as most of the groups of false course heads are not likely to occur in practical methods, it has not been attempted here.

*Note* 3: Group I. has been set out as if it consists of a single set; in fact, it consists of seven identical sets, since 2345678 has seven coursing pairs in plain course positions.

© Maurice Hodgson, 1962.

Reprint 1965.