PROFESSOR TAIT ON KNOTS. ei 
as in Plate XV. fig. 14, where the dotted parts may cross and recross in any 
conceivable way, but must begin and end as above. 
The number of ways in which such coils can be exhibited in plaits essen- 
tially distinct from one another is therefore, if m be the number of laps, 
m—1n—2....2.1., All the other possible arrangements, n—1 times the last 
written number, correspond to links or, at all events, to more than one continu- 
ous cord. 
§ 26. From this point of view another notation for clear coils may be given 
in the form 
ay Ba 
Bie te edi 
Here a, B, y.... are, as above, the several strings plaited, so that in the coil 
B is the prolongation of a, y that of 8, &c., and a that of the last of the series. 
. a e ° ° ° 
_ The expression 8 means that «a crosses over B. It is sometimes useful to indicate 
whether a crossing takes place to the right or left. This is done by putting 
+ or — over the symbol. Thus the four crossings above may be more fully 
written as 
+—+— 
ayBa 
Bre Sede cy 
The properties of this notation were examined in detail in my first paper; but 
as they are more curious than useful, I merely mention one or two. 
Thus the combination just written cannot be simplified in itself; but 
+——— i 
Lo a a eet ae? 4 
yeaa siemens Metab 
This notation requires care. For instance, the terms 
BB 
are simply nugatory, and may be cancelled. But, on the other hand, the terms 
a B 
Ba 
usually add to the beknottedness of the whole scheme. 
When the scheme is not compatible with a clear coil there occur terms of 
the form 
a 
a , 
and the application of this method becomes very troublesome. 
