PROFESSOR TAIT ON KNOTS. 163 
branch from the one point of view is outside it from the other, and vice versd. 
In fact, because the new figure is represented by the same scheme as the old, 
the numbers of sides of the various compartments are the same as before, and 
so also is the way in which they are joined by their corners. The deformation 
process is, in fact, simply one of Ayping, an excellent word, very inadequately 
represented by the nearest equivalent English phrase “turning outside in.” 
Hence to draw a scheme, select in it any closed circuit, ¢g., A ....A—the 
more extensive the better, provided it do not include any less extensive one. 
Draw this, and build upon it the rest of the scheme ; commencing always with 
the common point A, and passing each way from this to the neat occurring of 
the junctions named in the closed circuit. [It is sometimes better to construct 
both parts of the rest of the scheme znside,.and then invert one of them, as we 
_ thus avoid some puzzling ambiguities.| Inversions with respect to various 
origins will now give all possible forms of the scheme, though not necessarily 
of the knot. 
ns 16. Applying ehese methods to the “ trefoil ” Enby (§ 6) 
we easily see that if O be external, or be inside the inner ¢hree-sided compart- 
ment, we reproduce (generally with much distortion, but that is of no conse- 
quence, § 2) the same form; but if O be in any one of the two-sided compart- 
ments, we have the form 
This again is reproduced from itself if O be external, or be within either of 
the two-sided compartments. But it gives the trefoil knot if O be se plived inside 
either of the three-sided compartments. 
Here notice that the angles of the two-sided compartments are left-handed, 
and those of the three-sided right-handed in each of the figures. The perverted 
or right-handed form is of course 
and its solitary deformation is the perversion of the other figure above. 
