239 
of Edinburgh, Session 1876 - 77 . 
But in the latter scheme above, we have to deal with the curves 
A D B A and C E 0 E, and in the last of these we cannot have the 
junctions alternately - 4 - and — as required by our fundamental prin- 
ciple. In fact, the scheme would require the point C to lie simul- 
taneously inside and outside the closed circuit A D B A. 
Or we may treat ADB A and 0 E D C as closed curves inter- 
secting one another and yet having only one point, D, in common. 
(c) Thus, to test any arrangement, we may strike out from the 
whole scheme all the letters of any one closed part as A A , 
and the remaining letters must satisfy the fundamental principle. 
Or we may strike out all the letters of any two sets which begin 
and end similarly, e.g ., A...X,X...A, the two together 
being treated as one closed curve, and the test must still apply. 
More generally, we may take the sides of any closed polygon 
as A - X, X - Y, Y - Z, Z - A, and apply them in the same way. 
But in this, as in the simpler case just given, the sides must all 
be taken the same way round in the scheme itself. 
(d.) Such schemes as the latter of the two in (6) above may be 
made algebraically possible by slightly changing our assumptions. 
Thus, for instance, we might admit of a triple point, and agree not 
to reckon it as an intersection on a continuous oval provided one 
of the remaining branches goes into the oval, and the other comes 
out of it, these two not necessarily intersecting one another. In 
the case specified the triple point would be E supposed to lie on 
the oval ADB A, and not to be counted as an intersection while 
we pass round that oval. But this is a mere algebraic escape from 
a geometrical difficulty, and will not necessarily help us when we 
deal with knots on actual cords or wires. 
II. A possible scheme being thus made, with the requisite 
number of intersections, let it be constructed in cord, with the 
intersections as above alternately + and — . Then [since all schemes 
involving nugatory points, like those above mentioned, must be got 
rid of, as they do not really possess the requisite number of inter- 
sections] no deformation which the cord can suffer will reduce, 
though it may increase, the number of double points. If it do 
increase the number, the added terms will be of the nugatory 
character presently to be explained. If it do not increase that 
number, the scheme will in general still represent the altered 
