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Proceedings of the Royal Society 
undone, and 9 A j has been twisted into 9 A q, as 9 Ar 2 has into 9 A&, 
both by the twist of Tait, which is above analysed. In Tait’s twist 
the part rotated is not the same figure between P and E after rota- 
tion as before. 
There may, of course, be any number of crossings in the rotating 
part about PE. 
If here we consider P and E as both crossings, 3 and 4 are two 
unsolids of ten crossings, complementaries of each other about PE. 
When the knot to be twisted is such that the rotated part is the 
same after rotation as before, 3 has no complementary, and the 
result is simply one pair of convertibles, viz., K on the left instead 
of 9 A j, turned into K' on the right instead of 9 A qj, and not two pairs 
as here, 9 A j turned into 9 A q and 9 Ar 2 into 9 A b. 
If, in the above process, we have only two summits a and b 
between P and E in 3, making the 2-gon ab collateral with the tri- 
angles abV and ab E, we have the simplest possible case of twisting ; 
and in this the rotation makes no change, 4 and 3 being as projec- 
tions identical. 
