PROFESSOR TAIT ON KNOTS. 153 
and we shall denote by an appended numeral the number of times the opera- 
tion above has to be performed. Thus, in the example just given it will be 
found that 
ro=da ¢ 2 
Pre= th 0.2. 
§ 6. With one intersection or two only, a Anot is thus impossible, for the 
crossings must necessarily be nugatory. Hence we commence with three. 
And here there is but one case, for by our rule we must write A, B, C in the 
odd places, and we have no choice as to what to interpolate in the even ones. 
Thus the only knot with three intersections has the scheme 
ACBACBIA 
One of its two projections is the “ trefoil ” knot below. 
For four intersections our choice in the even places is restricted to C or D 
for the second, D or A for the fourth, &c., as expressed below, 
Coir AB 
De A B€., 
Now, if we take C to begin with, we obviously must take D next, else we shall 
| not get it at all. Similarly A must come third. And if we begin with D, we 
must end with C, so that this case also is determinate. The only possible sets, 
therefore, are given by these two rows as they are written. But it is obvious 
that, as they are in cyclical order, the full schemes will be identical if one be 
' read from the beginning, the other from the A in the even places. Thus they 
| represent the same arrangement, and the sole knot with four intersections has 
| the scheme 

ACRDCAD BY A 
One of its two projections is given by the annexed figure : -- 
§ 7. When we have jive intersections, our choice for the even places in order 
is limited to the following groups of three for each, viz. :— 
VOL. XXVIII. PART I. 2k 
