243 
of Edinburgh, Session 1876-77. 
Here a, /3, y . , . are, in order, the several strings plaited— so that 
in the coil (3 is the prolongation of a, y that of } 3 , &c., and a that of 
the last of the series. The expression ^ means that a crosses over 
(3. 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 H 1 — 
a y (3 a 
(3 ay/?'**’ 
The properties of this notation are examined in detail. It is 
shown, that the combination just written cannot be simplified in 
itself ; but that 
a y y a 
(3 a (3 (3 
VV,&C. 
p 
a 
This notation requires care. For instance, the terms 
a a 
(3(3 
are simply nugatory, and may be written off. But, on the other 
hand, the terms 
a (3 
(3 a 
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. 
(a.) When the scheme has no merely nugatory intersections, 
the most complete knotting is secured by alternate crossings above 
and below ; or, as we may write, 
A X B Y C &c. 
H + — + 
and here there is no continuation of sign. 
(b.) Cases in which there is no knot at all may be obtained for 
any scheme by making each letter positive on its first appearance. 
The various coils are then, as it were, paid out over one another. 
This process will give rise, in general, to but few changes of sign : 
2 K 
VOL. IX. 
