

174 PROFESSOR TAIT ON KNOTS. 
fig. 21, and the same looked at from the other side, 7.¢., with all the signs 
changed. 
Hence in the scheme | 
| Hp Ot ee stone 1 ec ae 
$+ Fo = = | 
(where the order is again indifferent in each of the groups) we can always leave 
out P and Q, unless R be negative and S positive, 7.¢., unless this part of the 
scheme has in itself the greatest possible number of changes of sign. 
But. when we can-thus strike out P and Q, it is necessary to observe that 
in RS or SR, which must occur at some other part of the scheme, the order is 
tobe changed. Thus 
Ce SPO ES @ J en 
: +++ —+- - = 
is simplified into 
TAG aoe ra, SU aes Bo) Ua a eee 
+ ~ - 
§ 30. Such a portion as that figured in Plate XV. fig. 22 evidently goes out 
of itself, whatever be the.character of B; 2.¢., the whole of it 
may be struck out of any scheme. In fact, whichever sign be given to B, § 29 
applies and removes two of the intersections. Then § 28 disposes of the 
remaining one. 
This is merely a particular case of the general and obvious theorem, that — 
any portion of a coil which may be treated as a separate coil, and which, if 
alone, could be reduced, may be reduced 27 situ. 
A more general theorem, which includes the preceding, is that, if in 
Bee bs 6 em Ctl hes a Seal 
the signs of B, C,...G, H, where they occur between the two A’s, are all 
alike, all these intersections, including A, may be struck out. This is quite 
obvious, because it indicates a complete turn of the coil entirely above or below 
the rest. When one or more of B, C, G, H has a different sign from nae others, 
a less amount of simplification is usually still possible. 
Along with this we may take the case of fig. 23. Here we have 
2 Q RP Sua ROaS oie ies 
——++4++4 —+— 
If the sign of P were changed these parts of the scheme would contain 
