PROFESSOR TAIT ON KNOTS. 175 
alternately + and —. The scheme obviously loses three intersections, and 
becomes ? 
Ets Qed 8) cepa ae 
If the signs in the complete knot, with the exception of that of P, were all 
originally + and — alternately, there will generally be farther reductions possible. 
§ 31. A glance shows that the first of the diagrams, 24, 25, Plate XV., can 
be reduced to the second. Hence in the scheme of a knot 
itera Dh PYOPREP Sle OUR ee 
++—— —+ 
may be simplified into 
ane Qu Rs .24h. si 2 R-Q 
+ — += 
| N.B.—-The essential point is that P and Q should have the same sign, and 
R the opposite. If Q and R had the same sign they might both be struck out 
§29. Butif P and Q have different signs, as also Q and R, no simplification 
can be effected, though, as has been shown in § 11,a Coes of scheme is practi- 
cable. ] 
§ 32. The scheme 
ERC eee AMN....PQG... 
+++ — 
_always admits of striking out A andG. But special consideration is necessary 
as to what is to take the place of B, C,...E, F. Their substitutes will all be 
positive, and may be called m, n,...p, gq, since they are in number the same 
as M, N,.... P, Q—irrespective altogether of the number of B, C,.... E, F. 
In fact, M and m, N and n,.... &c., lie (as near one another, in pairs, as we 
please) on the several turns of the coil which intersect the arc A M . 
QG. And m, n,....&c., are on the opposite side of that arc from B, C,....F 
§ 33. There are numberless other special rules, but those just given are 
among the simplest, and they are in general sufficient for coils with only a 
moderate number of intersections. With the present notation it is not easy to 
classify them, or to show how they may be exhibited as particular cases of 
more general rules. We will therefore, for the present, employ them only for 
the simplification (where possible) of a few diagrams of knots. But it must be 
particularly noticed that the simplifications above are mainly such as tend to 
remove continuations of sign from a scheme, none of them but the first being 
applicable to a scheme whose signs present no continuations. 
