

_ or beginning with A, 

PROFESSOR TAIT ON KNOTS. 151 
of letters intervenes between the two appearances of each of the remaining 
letters. 
Or we may strike out all the letters of any two sets which begin and end 
similarly, ¢.g., A... X, X... A, the two together being treated as one closed 
curve, and the on must still apply. 
More generally, we may take the sides of any closed polygon as A—X, 
X— Y, Y—Z, Z—A, and apply them in the same way. But in this, as in the 
simpler case just given, the sides must all be taken the same way round in the 
- scheme itself. 
_ A simple mode of applying these tests will he given later, when we are 
dealing with the question of Beknottedness. 
It may be well to explain here how a change of the crossing selected as 
the initial one alters the scheme. Take the simple case of making B the first, 
and reckoning on from it. Then B becomes A, &c., and the scheme, which 
may be any whatever, suppose for example 
AFBLILCEDAG... 
becomes (by writing for each letter that which alphabetically precedes it) 
NEAKBDCG... 
AKBDCG. 
Hence the letters 
F, L, E, / 7 ee 
in the even places . a scheme are edpswalens to 
K, D, G, a J 
1.é., we may change each to the preceding letter taken in the cyclical order of 
| the alphabet and put the first to the end, or vice versd, without altering the 
scheme. An arrangement of this kind is unique (reproducing itself) if the 
| letters are in cyclical order ; and if the number of letters be a prime, any arrange- 
| ment is either unique or is reproduced after a number of operations of this kind 
| equal to the number of letters. If it be not prime, arrangements may be found 
| which will reproduce themselves after a number of operations equal to any one 
| of its aliquot parts. 
Another lawful change is this :—Begin from the A in the even places anil 
| letter as usual, 7.¢., start from the same crossing as before, and in the same 
_ direction round the curve, but not by the same branch of the cord or wire. 
This will be evident from an example. Beginning at the second A, and letter- 
| ing alphabetically every second crossing, we have the suffixed letters. 
ADBACFDBECFE|A 
Poa BUN OO 
