PROFESSOR TAIT ON KNOTS. 173 
nor have I even discovered, what would probably solve this difficulty, any 
perfectly general method of pronouncing at once from an inspection of its 
scheme or otherwise, whether a knot is reducible or not. It is easy to give | 
multitudes of special conformations in which reduction can always be effected ; . 
but of these I shall give only a few, with the view of showing their general 
character. 
_ One very simple case of such reduction has Ehety been given, viz., where 
a letter occurs twice in ‘succession. 
For, if we have as part of a scheme, the wale 
-PQQR... 
the corresponding part of the coil must have the form shown in Plate XV. fig. 19. 
Whichever way the crossing at Q is effected, the loop can be at once got rid 
of, and it is thus nugatory, because ee scheme shows that it 1s. not intersected by 
any other branch. 
If we put in the signs of. the crossings, they must obviously be different for 
the two Q’s; and thus in 
| te Oy QR io uet « 
+ — 
we may treat them as + Q — Q = 0, and obliterate Q altogether. _ 
An immediate consequence of this is, of course, that any group such as 
RPORR OP. |< 
whatever be the number of letters arranged in this form, may be wholly struck 
out. .Cases corresponding to this have been already figured in § 1. 
§ 29. Another useful step in simplification occurs when we have a scheme 
Beataining the following terms :— 
for then both P and Q may be struck out. 
| V.B.—The order of P and Q need not be the same at each occurrence, the 
essential thing is that they should twice occur together, and with like signs. 
This explanation shows that the process. is not confined to clear coils. | 
For the corresponding part of the diagram must evidently be. of the form 
shown in Plate XV. fig. 20, since the scheme shows that there are no inter- 
sections between P and Q on either branch. Hence, as P and Q have the 
same sign for each branch, one branch may be slipped off from the other with- 
out otherwise altering the coil. 
If a single turn of the coil pass across hettredn P and Q, the only ways in 
which it can prevent the slipping off just described are that shown in Plate XV. 
VOL. XXVIII. PART I. 2Y 
